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University  of  California. 


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EXPERIMENTAL  STUDIES  IN  PSYCHOLOGY  AND  PEDAGOGY 


Editor: 
LIGHTNER  WITAIER 

UNIVERSITY   OF  PENNSYLVANIA 


III.     THE  APPLICATION  OF  STATISTICAL  METHODS  TO 
THE  PROBLEMS  OF  PSYCHOPHYSICS 


THE  APPIJCATION   OF   STATISTICAL 

METHODS  TO  THE  PROBLEMS 

OF  PSYCHOPHYSICS 


BY 

F.  M.  URBAN,  Ph.D. 

HARRISON    FELLOW    FOR   RESEARCH 
University  of  Pennsylvania 


FROM   THE  LABORATORY   OF    PSYCHOLOGY 


PHILADELPHIA,  PA. 

THE    PSYCHOLOGICAL    CLINIC    PRESS 
1908. 


-'<  f»,  ^ 


TOWN    PRINTING    CO. 
PHILADELPHIA 


Br£3  7 


PSYCH. 
LIBRARY 


PREFACE 

The  following  study  deals  with  the  methods  which  serve  for 
the  determination  of  the  threshold  of  difference.  These  methods 
can  be  easily  adapted  to  the  determination  of  the  maximum  and 
minimum  stimulation,  so  that  it  seemed  justified  to  refer  in  the 
title  of  the  book  to  the  problems  of  psychophysics  in  general. 
These  problems  are  treated  by  such  propositions  of, the  calculus  of 
probabilities  as  apply  to  the  results  of  observations  of  statistical 
numbers  of  relative  frequency.  The  term  "statistical  method" 
has  two  slightly  different  meanings.  The  first  refers  to  the  method 
of  collecting  observations  on  a  group  of  individuals  or  on  one  indi- 
vidual at  different  times.  The  data  obtained  in  this  way  must  be 
subjected  to  some  kind  of  treatment  and  the  term  "statistical 
method"  refers  in  its  second  meaning  to  every  algorithm  for  the 
numerical  evaluation  of  these  data  or  to  every  definite  method  l^y 
which  conclusions  can  be  drawn  from  them.  One  may  say  in  gen- 
eral that  a  problem  is  treated  statistically,  if  the  data  are  collected 
with  a  view  to  determining  certain  numbers  of  frequency  which 
serve  as  a  basis  for  further  deductions. 

Some  of  the  ideas  expounded  in  this  book  date  back  a  consid- 
erable time.  The  considerations  of  the  last  chapter  originated  in 
the  course  of  a  study  of  some  physical  phenomena  and  they  were  fol- 
lowed up  later  on  in  connection  with  astudy  of  thelawof  Gomperz- 
Makeham.  Most -of  the  formulae  of  the  third  chapter  have  been 
known  to  me  for  several  years,  and  the  idea  of  a  purely  formal 
treatment  of  the  results  of  psychological  experiments  came  to  me 
in  the  course  of  the  experiments  of  Mr.  Kobilecki,  which  were 
reported  lately  in  the  "Psychologische  Studien".  These  small 
ideas,  however,  would  have  remained  undeveloped  for  a  long  time, 
had  not  Prof.  Witmer  brought  to  my  knowledge  an  experimental 
arrangement  which  could  be  adapted  to  the  purposes  of  this  in- 
vestigation. It  is  so  extremely  easy  to  have  brilliant  ideas  in 
psychology  that  theoretical  discussions  justly  meet  with  a  certain 

v 


180105 


VI  PREFACE 

distrust,  if  they  are  not  supported  by  adequate  experimental  ma- 
terial. The  impossibility  of  deciding  on  an  experimental  pro- 
cedure which  would  yield  suitable  material  for  the  test  of  the  theo- 
retical deductions  was  the  chief  reason  for  not  publishing  this  an- 
alysis of  the  psychophysical  methods  before.  This  delay  of  more 
than  five  years  proved  to  be  a  very  fortunate  circumstance,  not 
only  because  the  growth  of  such  ideas  is  very  slow,  but  also  be- 
cause the  problems  of  psychophysics  have  been  the  subject  of 
several  important  publications  in  the  last  years.  Wundt,  Mliller, 
Lipps  and  Titchener,  have  treated  of  the  psychophysical  methods 
in  the  last  few  years;  especially  by  the  work  of  Titchener  is  the 
literature  on  this  problem  opened  up  as  it  was  never  before.  A 
similar  great  help  for  the  analysis  of  the  psychophysical  methods 
was  found  in  Czuber's  work  on  the  calculus  of  probabilities  and  in 
those  parts  of  the  "Encyklopadie  der  Mathematischen  Wissen- 
schaften"  which  deal  with  the  calculus  of  probabilities  and  its 
application  to  statistics.  Comparatively  little  reference  to  litera- 
ture is  made  in  this  book.  Constant  reference  to  work  previ- 
ously done  was  out  of  the  question,  because  it  would  have  made 
the  book  too  voluminous,  and  occasional  quotations  are  of  little 
or  no  use.  Add  to  this  that  the  present  investigation  has  not  the 
same  starting  point  as  the  treatises  of  the  previous  authors,  and 
that,  therefore,  it  would  have  been  necessary  to  argue  against 
views  which  in  themselves  are  interesting  and  of  merit.  This 
would  have  given  an  entirel}^  erroneous  impression  of  my  opinion 
of  the  value  of  these  views,  and  hence  it  seemed  best  to  describe 
in  separate  papers  the  different  phases  of  the  development  of  the 
psychophysical  methods.  In  so  far  as  the  English  literature  is 
concerned  this  presentation  will  be  given  in  one  of  the  reports  on 
"Die  Psychologie  in  Amerika",  which  appear  from  time  to  time 
in  the  Archiv  far  die  gesammte  Psi/chologie. 

In  working  out  the  numerical  results  it  was  made  a  rule  to  re- 
peat the  computation  independently  whenever  it  was  not  possible 
to  apply  a  thoroughgoing  check.  The  course  of  the  calculations 
is  given  in  great  detail.  This  was  done  in  order  to  illustrate  the 
theoretical  deductions  by  numerical  examples,  and  to  show  that 
the  actual  application  of  the  methods  described  here  is  simple 
and  that  it  is  shorter  than  the  methods  used  at  present.     This 


PREFACE  VII 

remark  refers  especially  to  the  method  of  just  perceptible  differ- 
ences, which  requires  four  times  as  many  experiments  as  the  new 
method  to  give  the  result  with  the  same  degree  o.'  accuracy. 
Great  care  was  taken  to  present  all  the  deductions  in  as  simple 
a  form  as  possible.  Mr.  Titchener's  work  has  set  a  standard  for 
such  treatment,  and  it  may  be  hoped  that  the  following  consid- 
erations will  be  intelligible  to  everyone  who  has  gone  through  the 
"Manual".  This  does  not  mean  t4iat  all  the  theorems  used  are 
spoken  of  in  this  work,  but  that  they  are  such  that  they  might  be 
understood  by  ever}'  one  who  can  read  Mr.  Titchener's  book.  This 
rule  of  avoiding  complicated  deductions  made  necessary  the  cur- 
tailing of  a  demonstration  in  the  third  chapter  at  the  place  where 
reference  is  made  to  Bruns's  theorem  of  the  conservation  of  the 
0(^)-type.  The  complete  solution  of  the  problem  would  have 
required  a  long  demonstration  of  a  very  technical  character,  in 
which  some  of  the  more  complicated  functions  are  used.  It 
therefore  seemed  best  to  give  it  at  some  other  place. 


TABLE  OF  CONTENTS. 

PAGE. 

Ch.^pter      I.     Description  of  the  Experiments i 

Chapter   II.     The   Statistical    Numbers   of    Relative 

Frequency    19 

Chapter  III.     On  The  Method  of  Just  Perceptible  Dif- 
ferences      40 

Chapter  IV.     The  Equality  Cases 99 

Chapter    V.     The  Psychometric  Functions     106 

Chapter  VI.     A  General  Inquiry  Concerning  the  Psy- 
chometric Functions     139 


PROBLEMS    OF  PSYGHOPEYSrCS 


CHAPTER  I. 

DESCRIPTION   OF   THE   EXPERIMENTS. 

The  results  of  a  series  of  experiments  on  lifted  weights  are  the 
basis  of  our  discussion.  The  weights  were  hollow  brass  cylin- 
ders 7.5  cm.  in  diameter  and  3  cm.  in  height,  closed  at  one  end. 
The  weight  of  the  metal  of  the  cylinders  as  they  came  from  the 
shop  was  slightly  less  than  the  smallest  weight  to  be  used:  by 
pouring  different  quantities  of  melted  paraffine  into  the  cylin- 
ders sufficient  to  make  each  cylinder  a  little  heavier  than  de- 
sired and  then  scraping  out  small  quantities  of  the  hardened 
paraffine,  the  weight  could  be  adjusted  very  exactly.  The  ex- 
actitude of  this  adjustment  finds  a  limit  only  in  the  exactitude 
of  the  balance  used.  The  variations  which  occur  in  weights 
prepared  in  this  way  are  inconsiderable;  they  were  observed 
regularly  and  Table  1  (Appendix  p.  173)  gives  the  variations 
which  occurred  in  winter  and  spring  1907.  The  first  column 
of  this  table  gives  the  number  by  which  each  weight  could  be 
identified.  Since  the  size  and  the  external  appearance  of  all 
the  weights  was  the  same,  it  was  necessary  to  distinguish  them 
in  some  way;  this  was  accomplished  by  stamping  small  numbers 
in  the  centre  of  the  upper  side  of  every  cylinder.  These  num- 
bers could  be  seen  only  on  close  inspection.  The  second  column 
of  Table  1  gives  the  weight  in  grams  to  which  the  c^dinders  were 
adjusted  before  the  experiments  were  started;  this  adjustment 
as  well  as  the  other  observations  was  made  with  a  balance 
which  showed  differences  of  1  mgr.  The  differences  between 
the  observed  weight  and  the  adjusted  weight  are  given  in 
milligrams  for  every  day  of  control,  the  dates  of  which  are 
given  at  the  heads  of  the  columns.     The  plus  sign  indicates  an 

1 


2  miOBLEMS  OF  PSYCHOPHVSICS 

increase  of  weight,  the  minus  sign  a  loss  of  weight.  Weights 
in  which  a  positive  or  negative  difference  of  more  than  10  mgr. 
from  the  standard  was  observed,  were  readjusted;  this,  however, 
happened  only  three  times  in  the  period  of  three  months  for 
which  the  results  are  reported.  The  last  column  of  the  table 
gives  the  sum  of  all  the  variations,  understanding  by  variations 
the  positive  or  negative  difference  which  the  weight  has  under- 
gone between  two  observations.  The  cylinder  No.  3,  for  instance, 
was  adjusted  for  the  weight  of  100  gr.  and  the  observed  differ- 
ences from  this  standard  are  given  in  Table  1  as  -3,  3,  -5  mgr. 
for  the  first  three  observations.  This  means  that  on  these  days 
the  weight  of  this  cylinder  was  found  to  be  99.997,  100.003  and 
99.995  gr.  We  have,  therefore,  an  increase  of  6  mgr.  between 
the  first  and  the  second  observation,  and  a  decrease  of  8  mgr. 
between  the  second  and  the  third.  The  sum  of  all  the  varia- 
tions does  not  exceed  31  mgr.  for  anyone  of  the  weights  and  on  the 
average  it  is  less  than  17  mgr.  A  glance  at  the  table  shows 
that  the  deviations  from  the  adjusted  weight  are  negative  in 
the  majority  of  cases,  but  it  is  worth  while  noticing  that  the 
variations  between  the  single  observations  were  negative  in  46 
eases  and  positive  in  45  cases,  whereas  in  23  cases  no  variations 
could  be  detected.  The  original  adjustment  of  the  weights 
was  made  in  March,  1906.  It  seems  that  the  cylinders  suffer 
a  loss  of  weight  in  the  beginning,  but  that  they  undergo  little 
variation  afterwards.  The  smallness  of  these  variations  is 
chiefly  due  to  the  use  of  anhygroscopic  materials,  as  may  be 
seen  by  a  comparison  with  the  variations  of  weights  which  are 
made  of  materials  more  susceptible  to  hygroscopic  influences. 
For  the  sake  of  this  comparison  Table  2  (Appendix  p.  173)  is 
given  which  contains  similar  observations  for  four  weights  of 
two  patterns.  The  weights  which  are  called  in  this  table  I  and 
II  are  Cattell  weights  and  belong  to  the  set  of  weights  which 
were  used  by  Fullerton  and  Cattell  in  the  series  of  experi- 
ments which  are  discussed  in  the  publication  of  these  authors.* 
Weights  III  and  IV  are  solid  wooden  blocks  filled  with  lead 
for  adjustment.     The  table  shows  the  variations  of  these  four 

*FuLLERTON  and  Cattell,  The  Perception  of  Smill  Difjcrences,   1S9-',  pp. 
116-129. 


DESCRIPTION  OF  THE  KXPEHIMEXTS  3 

weiiihts  in  the  period  from  April  26  till  May  10;  the  upper  part 
of  the  table  gives  the  results  of  the  observations  in  grams  and 
the  lower  part  of  the  table  gives  the  variations  between  two 
observations.  It  may  be  seen  from  the  table  that  the  varia- 
tions of  the  wooden  blocks  are  by  far  the  largest;  they  are  two 
or  three  times  as  large  as  those  of  the  Cattell  weights.  These 
weights  are  not  fit  for  experimentation  which  aims  at  any  ac- 
curacy not  only  because  the  variations  are  ver}'  large,  but  also 
because  it  is  difficult  to  correct  a  negative  variation.  The  vari- 
ations of  the  two  Cattell  weights  are  considerably  smaller,  but 
the  variations  of  one  weight  in  three  weeks  are  almost  twice 
as  great  as  the  variations  of  all  our  IS  weights  in  the  course  of 
a  little  less  than  three  months.  It  need  not  be  remarked  that 
all  the  weights  were  kept  under  the  same  conditions,  the  disad- 
vantage being  on  the  side  of  our  weights  which  were  exposed 
to  rough  handling  in  the  course  of  the  experimentation. 

These  cylinders  were  not  only  exactly  alike  to  sight  but  also, 
what  is  perhaps  more  important,  to  touch.'''  The  metallic  sur- 
face of  the  weights  was  polished  so  that  it  gave  the  same  impres- 
sion of  smoothness  no  matter  where  it  was  touched.  Since  all 
the  weights,  furthermore,  were  of  the  same  metal  and  care  was 
taken  ta  keep  them  under  the  same  conditions  of  temperature 
(by  keeping  all  the  weights  in  the  same  room  and  by  not  touch- 
ing with  the  warm  hand  one  weight  more  frequently  than  the 
others)  an  influence  of  differences  of  temperature  was  avoided. 
The  only  quality  by  which  one  cylinder  differed  from  another 
was  its  weight. 

These  weights  were  arranged  along  the  circumference  of  a 
round  turning  table  at  regular  intervals,  the  position  of  each  weight 

*The  sensations  of  touch  are  an  important  component  of  the  sensation  of 
weight,  and  for  this  reason  it  is  necessary  to  keep  them  constant,  if  the  pur- 
pose of  the  experiments  is  an  analysis  of  the  sensations  of  weight.  The  im- 
portance of  the  sensation  of  touch  for  the  judgment  of  the  weight  of  a  body 
was  recognized  by  Fechner,  Elemente  der  Psychophysik,  \'ol.  I,  p.  19V1, 
and  this  observation  was  confirmed  lately  by  A.  Lehm.\nn,  Beiirdge  zur 
Psychodynamik  der  Gewichtsempfindungen,  Arch.  f.  d.  ges.  Psycliologie,  Vol. 
6,  1906,  pp.  446-448,  who  found  that  variations  of  the  sensations  of  touch  may 
cause  differences  in  the  estimation  of  weights,  which  may  be  equal  to 
those  caused  by  differences  of  {  of  the  intensity  of  the  stimulus. 


4  PROBLEMS  OF  PSYCHOPHYSICS 

being  determined  by  one  of  the  numbers  1-14;  a  great  number 
of  variations  in  the  order  of  the  stimuli  may  be  obtained  by 
interchanging  the  position  of  these  14  weights.  A  general  feat- 
ure of  all  the  arrangements  used  in  our  experiments  was  that 
weights  of  100  gr.  were  placed  at  the  odd  numljers.  The  weights 
placed  at  the  even  numbers  had  to  be  compared  with  these 
standard  weights.  The  weights  Avere  always  lifted  with  the 
right  hand.  The  right  forearm  of  the  subject  rested  on  a  firm 
table  in  such  a  position  that  the  hand  from  the  wrist  extended 
over  the  edge.  The  subject  was  sitting  at  this  table  which  was 
placed  so  that  the  hand  of  the  subject  was  vertically  over  one 
of  the  weights  of  the  turning  table.  The  turning  table  with 
the  weights  on  it  was  shut  off  from  the  view  of  the  subject  by 
means  of  a  screen.  The  height  of  the  turning  talkie  could  be 
regulated  so  that  the  weights  were  within  easy  reach  of  the  sub- 
ject. The  lifting  was  done  entirely  from  the  wrist  and  no  part 
of  the  arm  above  this  joint  was  moved.  The  height  of  lifting 
was  not  regulated,  but  the  subject  was  instructed  to  lift  the 
weights  in  such  a  way  as  seemed  best  in  order  to  obtain  an  ac- 
curate judgment.  This  was  not  an  equivocal  instruction,  since 
every  subject  was  given  a  large  amount  of  practice  before  the 
actual  experiments  were  begun,  so  that  the  subject  could  find 
the  way  of  lifting  which  suited  him  best.  The  speed  of  the  move- 
ments of  the  hand  did  not  seem  to  vary  much  after  the  subject 
had  settled  down  to  a  certain  way  of  lifting.  Most  subjects 
favored  an  excursion  of  the  hand  of  approximately  5-7  cm.;  one 
subject  preferred  to  lift  his  hand  considerably  higher,  and  an- 
other adopted  a  very  small  excursion  of  2-3  cm.  This  is  in  so 
far  of  importance  as  the  time  of  the  movements  was  kept  con- 
stant and  the  excursion  of  the  hand  gives  an  indication  of  the 
velocity  with  which  the  weights  were  lifted.  The  height  of 
lifting  depends  to  some  extent  on  the  flexibility  of  the  hand  at 
the  wrist. 

The  movements  of  the  hand  were  regulated  by  a  metronome 
of  Malzel,  which  beat  92  times  a  minute  and  every  fourth  beat 
of  which  was  marked  by  the  stroke  of  the  bell.  On  the  stroke 
of  the  bell  the  hand  of  the  subject  went  down  and  grasped  the 
weight;   on  the  second  beat  the  hand  was  lifted;  on  the  third 


DESCRIPTION  OF  THE  EXPERIMENTS  5 

beat  the  weight  was  put  back  on  the  table,  and  on  the  fourth 
beat  the  hand  returned  to  its  original  position.  After  the  weight 
was  put  l)ack  the  operator  had  sufficient  time  to  turn  the  table 
and  to  bring  the  next  weight  directly  below  the  hand  of  the  sub- 
ject. The  experiments  thus  could  go  on  without  interruption 
and  were  continued  for  five  complete  turns  of  the  table,  so  that 
in  such  a  series  five  judgments  were  given  on  every  pair  of  com- 
parison weights.  The  position  of  the  hand  of  the  subject  was 
the  same  in  lifting  all  the  weights. 

The  judgments  referred  to  the  second  weight  of  the  pair.  The 
possible  judgments  are  that  the  second  weight  is  heavier  or  lighter 
than  the  standard,  or  that  Ijoth  weights  are  equal.  The  sub- 
ject was  allowed  to  express  the  degree  of  his  confidence  by  an- 
necting  one  of  the  numerals  1,  2,  or  3  to  his  judgment.  The 
unit  indicated  an  ordinary  degree  of  confidence  and  was  omitted^ 
2  and  3  indicated  higher  and  highest  degrees  of  confidence,  but 
it  was  found  that  the  subjects  rarely  went  beyond  2.  For  the 
so-called  doubtful  and  equality  cases  the  following  arrange- 
ment was  made.  It  is  a  fact  that  those  cases  where  the  judg- 
ment is  absolutely  doubtful  diminish  in  number  when  the  practice 
of  the  subject  is  increased.*  These  cases  alone  ought  to  be 
called  doubtful  cases,  because  in  the  other  class  of  cases  the 
judgment  is  "equal"  and  the  judgment  is  not  doubtful  at  all. 
Cases  where  the  subject  was  unable  to  form  a  judgment  occurred 
in  the  preliminary  series,  but  they  were  rare  in  the  actual  inves- 
tigation and  the  experiments  were  repeated  whenever  this  hap- 
pened. When  no  difference  could  be  detected  the  subject  was 
instructed  to  make  a  guess  whether  the  second  weight  was  heavier 
or  lighter,  but  to  mark  these  cases  by  the  words  "heavier  guess" 
or  "lighter  guess."  The  "guess "-judgments  are  theoretically^ 
i.  e.  by  their  definition,  identical  with  the  equality  cases,  but 
practically  we  met  with  some  difficulties.  Most  subjects  had 
in  the  beginning  of  their  training  some  difficulty  in  distinguish- 
ing  the    "guess "-judgments   from   judgments    which   expressed 

*Cfr.  WuNDT,  Physiol.  Psychologic,  5  ed.,  1903,  Vol.  1,  p.  482;  G.  E. 
MvELLEK  and  Fr.  Schumann,  Vber  die  psychologischen  Grundlagen  der  Ver 
gkichung  gchobener  Gewichie,  Arch..},  d.  ges.  Physiologic.     Vol.  45,  1889,  p.  40 


6 


PKOBLEM.S  OF  PSYCHOPHYSICS 


the  recognition  of  an  existing  difference  with  a  very  low  degree 
of  certainty.  Later  on  all  the  subjects  but  one  (subject  III) 
found  no  difficulty  in  separating  these  two  classes  of  judgments. 
This  subject  almost  never  seemed  to  have  the  impression  of 
equality  of  the  stimuli  and  he  consistently  gave  "guess "-judg- 
ments when  he  did  not  feel  quite  sure;  in  these  cases  he  mostly 
gave  the  judgment  "heavier  guess".  Since  it  was  not  possible 
to  eradicate  this  habit  by  a  very  considerable  amount  of  prac- 
tice given  to  this  subject,  it  was  decided  not  to  interfere  with 
his  way  of  judging. 

All  the  judgments  given  were  recorded.  For  this  purpose 
special  record  sheets  were  prepared  which  contained  seven  rows 
of  five  places  each.  Every  comparison  weight  was  given  one 
line,  so  that  the  35  experiments  of  running  the  table  around 
five  times  could  be  recorded  on  one  blank.  If  the  comparison 
weights  are  written  down  in  the  same  order  in  which  they  are 
placed  on  the  table,  every  column  of  the  record  gives  the  exper- 
iments of  one  turn  of  the  table  and  the  rows  contain  the  judg- 
ments given  on  one  comparison  weight  in  the  five  turns  of  the 
table.     The   following  example    will    make    clear    this   way    of 


104 
92 

h 

h 

h, 

h 

h 

5 

h 

hg 

1 

1 

hg 

3 

108 

h^ 

h 

h 

h 

h 

5 

88 

1 

hg 

1 

1 

1 

1 

96 

h 

h 

h 

hg 

h 
h 

5 

100 

hg 

h 

h 

h, 

5 

84 

hg 

1 

1 

1 

1 

1 

6 

6 

4 

4 

5 

keeping  the  records.     The  letters  h   and  1  stand  for  the  judg- 
ments "heavier"  and  "lighter."     The  "guess "-judgments  were 


DESCRIPTION  OF  THE  EXPERIMENTS  7 

recorded  by  hg  and  Ig,  according  to  whether  the  judgment 
"heavier  guess"  or  "lighter  guess"  was  given.  The  first  cohimn 
of  the  record  gives  the  comparison  weight;  it  is  not  necessary 
to  put  down  the  standard  weight  because  it  is  the  same  for  all 
the  pairs.  In  the  35  spaces  of  such  a  table  of  five  columns 
all  the  judgments  given  during  the  five  turns  of  the  taljle  may 
be  recorded  by  entering  everv  judgment  as  it  is  given  in  its  place 
running  from  the  top  of  the  columns  downward.  The  numbers 
in  the  last  column  refer  to  the  number  of  "heavier "-judgments 
given  on  this  weight,  no  discrimination  being  made  between 
"hg"  and  simple  "h "-judgments.  The  numbers  at  the  bot- 
tom of  the  columns  do  the  same  for  the  columns  i.  e.  they  give 
the  number  of  "heavier "-judgments  given  in  one  turn  of  the 
table  without  distinguishing  between  guesses  and  ordinary  judg- 
ments. 

This  experimental  arrancement  made  it  necessary  to  have 
three  persons  present:  The  subject,  the  operator  of  the  table 
and  the  recorder.  The  recorder  of  course  became  acquainted 
with  the  arrangement  of  the  comparison  stimuli,  and  it  was  found 
very  soon  that  this  knowledge  was  of  disturbing  influence,  if  the 
recorder  had  to  serve  as  subject  in  the  series  where  the  same 
arrangement  was  used.  For  this  reason  nol:)ody  was  used  as 
recorder  in  a  series  before  he  had  finished  it  as  a  sui^ject;  as  a  rule 
the  recording  was  done  by  the  conductor  of  the  experiments, 
who  could  not  avoid  having  some  knowledge  of  the  order  of  the 
comparison  weights  anyhow,  or  by  an  assistant  who  did  not  act 
as  a  subject.  In  this  way  it  was  tried  to  keep  from  the  subject 
all  knowledge  about  the  actual  relation  of  the  stimuli,  and  the  ex- 
perimental arrangement  fairly  excludes  the  possibility  of  the 
subject's  unwittingly  learning  what  this  relation  is.  Even  the 
knowledge  of  the  order  in  which  the  pairs  are  presented  does 
not  interfere  with  the  proper  performance  of  the  experiments, 
if  attention  is  devoted  to  the  experiments  and  not  to  the  order 
of  the  succession  of  the  judgments.  There  were  some  cases — 
very  few  in  number — where  the  subject  reported  that  the  idea 
was  impressed  upon  him  he  knew  the  order  of  the  pairs.  Cases 
of  this  type  were  investigated  on  the  spot  and  it  was  found  that 
this  impression  of  the  subject  was  only  seldom  justified.     These 


8  PROBLEMS  OF  PSYCHOPHYSICS 

cases,  as  a  matter  of  fact,  are  perhaps  more  interesting  as  ex- 
amples of  memory  illusions  and  of  the  origin  of  faulty  impres- 
sions based  on  incomplete  observations.  After  the  series  re- 
produced above,  for  example,  the  subject  reported  having  been 
conscious  that  there  was  in  the  series  a  succession  of  weights 
described  by  the  judgments  ho,  1,  hg,  and  the  subject  offered 
to  pick  out  these  weights.  A  number  of  trials  was  immediately 
given,  but  the  subject  could  not  make  up  his  mind  and  finally 
gave  up  this  task  as  impossible.  Then  the  record  of  this  series 
was  surveyed,  and  it  was  found  that  a  succession  of  judgments 
as  described  by  the  subject  did  not  occur  a  single  time  and  that 
the  hj-judgments  were  given  on  different  weights.  This  exam- 
ple points  out  two  facts:  1,  that  memory  is  very  unreliable  for 
events  on  which  attention  is  not  fixed  and  2,  that  ^he  degree 
of  confidence  with  which  a  judgment  is  given  does  not  depend 
solely  on  the  amount  of  objective  difference,  because  the  degree 
of  confidence  with  which  the  judgments  on  a  certain  pair  of 
comparison  weights  are  given  varies  considerably.  The  latter 
fact  is  well  illustrated  by  a  few  cases  where  hj  and  lo-judg- 
ments  were  given  on  the  same  pair  of  comparison  weights  inside 
the  same  series.     These  cases  were 


Subject  St.,  April  2,  1906,  I  A,  No.  8  (h  2  and  1  j  on  96) ; 
Subject  W.,  June  8,  1906,  IV  A,  No.  4  (h  ^  and  1 2  on  100) ; 
Subject  B.,  Oct.  23,  1906,  I,  No.  2  (h  2  and  1 2  on  108). 


Great  confidence  or  lack  of  confidence  are  no  criteria  of  the 
reliability  of  the  judgment,  and  it  is  not  likely  that  the  con- 
fidence in  the  judgment  is  in  a  simple  relation  to  the  difference 
between  the  stimuli.  There  are  cases  on  record  where  the 
subject  complained  of  very  little  or  no  confidence  at  all,  and 
where  a  survey  of  the  results  shows  that  the  judgments  in  this 
series  have  fewer  mistakes  than  any  other  series  of  the  same 
day.     In  some  cases,  on  the   contrary,  this  lack  of    confidence 


DESCRIPTION  OF  THE  EXPERIMENTS  9 

was  in  so  far  justified  as  the  results  showed  a  considerable  number 
of  erroneous  judgments.* 

There  are  two  points  in  our  experimental  procedure  which  need 
some  explanation;  the  first  is  the  regulation  of  the  movement 
of  the  hand  by  the  metronome,  the  second  is  the  taking  of  the 
experiments  in  series  of  35.  The  regular  sound  stimuli  may  have 
some  influence  on  the  judgment,  and  the  length  of  one  series  may 
tend  to  produce  states  of  fatigue  or  variations  of  attention  which 
make  results  of  different  parts  of  one  series  not  directly  com- 
parable. It  seems  that  only  the  second  possibility  requires 
special  attention,  l)ecause  the  use  of  the  metronome  for  regulating 
the  movements  of  the  hand  has  been  found  advantageous  and 
unobjectionable   in   several   previous  investigations.     It   will   be 

shown  in  the  second  chapter  that  there  exists    no  evidence  of  an 

...  X 

appreciable  influence  of  making  the  experiments  in  series  of  35. 
There  are  several  ways  in  which  the  beats  of  the  metronome  might 
possibly  influence  the  judgments: 

(l.)  Regular  acoustical  stimuli  produce  a  feeling  processf 
which  may  be  the  immediate  cause  for  judgments  in  a  certain  di- 
rection. It  is  very  easy  to  observe  this  waving  up  and  down  of 
feelings  when  listening  to  the  beats  of  a  metronome,  but  these 
feelings  are  by  far  less  noticeable  if  attention  is  primarily  con- 
centrated on  the  performance  of  the  experiments. 

(2.)  The  beats  of  the  metronome  may  influence  attention,  which 
is  the  psychical  function  affected  by  the  rythmic  motion  of  lift- 
ing the  weights. 

*rhe  first  part  of  this  observation  contradicts  the  statement  of  Fuller- 
ton  and  Cattell  that  with  an  increase  of  the  difference  of  the  intensities  of 
the  stimuli  our  judgment  changes  from  complete  uncertainty  to  confidence. 
Judgments  on  the  same  amount  of  objective  difi'erence  are  given  with  a  very 
variable  degree  of  confidence.  Of  course  it  is  possible  to  say  that  on  the  aver- 
age our  confidence  increases  with  the  difference  of  the  intensities,  but  taking 
such  an  ayerage  is  not  a  well  defined  process.  The  latter  part  of  the  obser- 
vation agrees  with  a  remark  of  these  investigators  {On  Ike  Perception  of 
Small  Difjerences,  1892,  p.  126  sqs.)  in  so  far  as  it  shows  that  accuracy  is 
not  necessarily  proportional  to  subjective  certainty,  an  observation  to  which 
these  authors  give  the  piquant  turn  that  those  subjects  who  are  most  confi- 
dent are  the  least  accurate  of  all. 

jW.  WuNDT,  Phys.  Psych.  5  ed.,  1903,  Vol.  3,  p.  23. 


10  PUOHLEMS  OF  PSYC'HOPH Y.SICS 

(3.)  The  Jicousticul  stimuli  may  interfere  with  the  mechanical 
contraction  of  the  muscles  in  such  a  way  as  to  reinforce  or  in- 
hibit the  action  of  the  muscles.  The  observations  of  Bowditch 
and  Warren/'^  Cleghorn,t  HofbauerJ,  Yerkes^f  and  others  show 
that  the  effect  of  two  stimuli  which  are  applied  in  not  too 
great  an  interval  is  never  equal  to  the  sum  of  the  effects  of  the 
stimuli,  but  that  they  interfere  in  such  a  way  as  to  inhibit  or 
reinforce  each  other.  There  is  little  doubt  that  there  is  the 
possibility  of  such  an  influence  of  the  regular  acoustical  stimuli 
on  the  contraction  of  the  muscles,  although  one  can  not  say  at 
present  what  effect  on  the  judgment  of  weight  such  an  interfer- 
ence of  stimuli  may  have. 

We  have  more  positive  information  in  regard  to  the  second 
point,  which  was  the  object  of  an  experimental  investigation 
of  Smith. §  For  our  present  purpose,  however,  it  is  less  im- 
portant to  have  an  analysis  of  the  factors  which  decide  the  judg- 
ment on  the  weight  of  a  body,  than  to  arrange  the  experiments 
in  such  a  way  that  the  conditions  for  all  the  judgments  are  the 
same,  no  experiment  being  favored  by  an  influence  which  is  not 
at  work  in  the  others  too. 

Similar  considerations  hold  good  for  the  possibility  of  an  in- 
fluence of  the  length  of  a  single  series.  Besides  fatigue  and  shift- 
ing of  attention  there  is  the  possibility  that  the  stimulation  pro- 
duced by  the  first  lifting  does  not  die  out  immediately,  but  tliat 
it  interferes  with  the  next  stimulation,  so  as  to  inhibit  it  or  rein- 
force it.  This  possibility  was  considered  recently  by  Lehmann.  || 
This  author  supposes  that  such  an  influence  exists  and  he  gives 

*H.  P.  Bowditch  and  J.  W.  Warren,  The  Knee-jerk  and  its  Physiologi- 
cal Modifications,  Journ.  of  Physiology,  Vol.  9,  p.  60,  1890. 

fA.  CuEGHORN,  The  Reinforcement  of  Voluntary  Muscular  Contractions, 
Amcr.  Journ.  of  Physiology,  Vol.  1,  p.  336,  1898. 

XL-  HoFBAUER,  Interferenz  zwischen  verschiedenen  Impulsen  im  Cen- 
tralnervensystem,  Arch.  f.  d.  ges.  Physiologie,  1897,  Vol.  68,  p.  546. 

"jR.  M.  YerkeS,  Bahnung  und  Hemmung  auf  tactile  Reize  durch  akustische 
Reize  beim  Frosche,  Arch.  f.  d.  ges.  Physiologie,  Vol.   107,  1905. 

§Margaret  Keiver  Smith,  Rythmus  und  Arbeit,  Phil.  Stud.  Vol.  16, 
1900,  pp.  71-133,  197-305. 

II A.  Lehm.\nn,  Beitrdge  zur  Psychodynamik  der  Gcwichtsempjindungen, 
Arch.  f.  d.  ges.  Psyclwlogie  Vol.  6,  1906,   pp.  425-499 


DESCRIPTION  OF  THE  EXPERIMENTS  1  1 

a  mathematical  expression  for  it.  An  influence  of  the  type 
Lehmann  describes  \vill  not  impair  the  experiments  Init  it  will 
go  in  as  a  constant  factor.  Indeed  since  the  interval  between 
two  successive  stimulations  is  constant,  no  matter  whether 
the  preceding  stimulus  inhibits  or  reinforces  the  following,  this 
influence  will  tend  to  produce  a  state  where  the  amount  of  in- 
hil)ition  or  reinforcement  is  the  same  for  all  the  weights.  This 
influence,  therefore,  will  be  distributed  equally  over  all  parts  of 
the  series  of  experiments. 

The  first  weight  in  every  pair  was  100  gr.;  it  was  compared 
with  weights  of  84,  88,  92,  96,  100,  104,  108  gr.  in  those  series 
the  results  of  which  are  discussed  here.  It  will  be  noticed  that 
the  series  of  comparison  weights  is  not  equally  extended  above 
and  below  the  standard.  The  motive  for  the  choice  of  this  series 
of  comparison  weights  was  that  it  was  observed  in  the  preliminary 
experiments  that  all  the  subjects  had  the  tendency  to  give  a  ma- 
jority of  "heavier"-judgments  on  the  comparison  of  100  with 
100.  It  does  not  seem  profitable  to  use  an  equal  number  of  posi- 
tive and  negative  differences,  because  one  would  get  a  very  great 
percentage  of  "heavier"-judgments  in  the  upper  part  of  the 
series  and  a  small  percentage  of  "lighter "-jud,2;ments  in  the 
lower  part,  or  one  would  suffer  from  the  opposite  inconvenience 
as  the  case  may  be  one  of  overestimation  or  underestimation  of 
the  second  stimulus.  It  came  out  as  a  result  of  our  experiments 
that  for  some  piu'poses,  e.  g.  for  the  determination  of  the  thres- 
hold by  the  method  just  perceptible  differences, the  use  of  at  least 
one  weight  beyond  108  gr.is  desirable,  and  that  with  some  subjects 
the  use  of  a  weight  as  far  down  as  84  ma}'  be  dispensed  with. 

Presenting  these  seven  pairs  of  stimuli  five  times  to  the  sub- 
ject in  the  way  described  takes  approximately  three  minutes 
(3'S").  After  the  series  was  finished  the  subject  was  given  a 
rest  of  some  five  minutes  before  another  series  of  experiments 
was  begun.  This  arrangement  enabled  us  to  make  six  or  seven 
series  of  experiments  in  the  course  of  one  hour.  It  was,  there- 
fore, possible  to  obtain  in  one  hour  of  experimentation  30  to  35 
judgments  on  each  one  of  the  pairs.  The  order  in  which  the 
pairs  were  presented  was  not  changed  before  100  judgments  on 
every   difference   were   obtained.     Such   a  group  of   100  experi- 


12  PROBLEMS  OF  PSYCHOPHYSICS 

ments  was  marked  with  a  Roman  numeral.  It  furthermore  was 
found  convenient  to  divide  the  results  in  groups  of  50  experiments, 
which  were  marked  by  the  corresponding  Roman  numerals  and 
by  the  Arabic  numerals  1  and  2;  III2,  thus,  refers  to  the  group 
of  experiments  which  comprises  those  from  the  51st  to  the  100th 
experiment  inclusive  in  the  third  series. 

The  data  of  five  different  series  were  used  for  the  present  in- 
vestigation. Two  of  these  series  were  made  in  spring  1906,  the 
others  in  the  year  1906-07.     The  order  of  the  pairs  was  this: 

la  96,  104,  108,  84,  92,  .  100,  88 

IVa  104,  92,  108,  88,  96,  100,  84 

I  96,  104,  108,  84,  92,  100,       88 

III  84,  104,  96,  100,  92,  108,       88 

IV  84,  88,  92,  96,  100,  104,  108. 

The  series  made  in  spring  1906  are  marked  by  the  letter  a.* 
Nineteen  subjects  were  experimented  on,  but  only  the  results 
of  those  are  used  for  the  puspose  of  the  present  study  who  have 
gone  through  series  I,  III  and  IV.  The  number  of  these  sub- 
jects was  seven,  three  of  whom  also  made  series  la,  and  IVa. 
The  subjects  will  be  spoken  of  as  subjects  I,  II,  III,  IV,  V,  VI, 
VII.  With  the  exception  of  subject  V  all  were  males.  Sub- 
ject III  was  42  years;  the  age  of  the  others  was  between  21  and 
30.  Subjects  I,  II  and  III  have  gone  through  all  the  series  and 
we  have  for  each  one  of  them  450  judgments  on  every  pair  of 
comparison  weights,  since  only  the  second  part  of  series  la  was 
available. t  The  other  subjects  gave  300  judgments  on  every 
pair.  The  total  number  of  experiments  for  each  one  of  the  first 
three  subjects  is  3,150,  and  for  each  one  of  the  others  2,100,  so 
that  our  discussions  are  based  on  the  results  of  17,850  experi- 
ments. This  number  is  great,  but  by  no  means  excessive;  it 
will  be  seen  that  it  is  sufficient  for  most  purposes. 

*The  plan  of  these  series  of  experiments  was  the  outcome  of  several  pre- 
liminary investigations  undertaken  by  the  Psychological  Laboratory  of  the 
University  of  Pennsylvania.  They  were  conducted  by  Professor  Witmer, 
from  whom  the  records  were  obtained  for'use  in  this  investigation. 

fThe  results  of  lal  show  the  influence  of  practice  and  were,  therefore,  no  t 
used  for  the  purpose  of  this  study.  Series  Ila,  Ilia  and  II  contain  experi- 
ments which  are  not  comparable  to  those  of  the  other  series,  because  the 
experimental  conditions  (time  of  lifting,  interval  between  the  pairs,  order  of 
standard  and  comparison  weight)  were  different. 


DESCRIPTION  OF  THE  EXPERIMENTS  13 

It  is  easy  to  see  that  the  arraa,ii;einent  of  our  experiments  is 
a  modification  of  the  one  described  by  Martin*  and  more 
recently  used  also  by  Smith,t  the  modifications  having  the 
purpose  of  eliminating  the  space  error.  The  principles  for  the 
choice  of  this  experimental  arrangement  were  laid  down  by  Miil- 
ler  and  Schumann,  whom  it  seemed  safe  to  follow  in  this 
respect.  The  difference  of  the  arrangement  used  by  Martin 
and  by  Smith  from  the  one  used  in  our  experiments  consists 
chiefly  in  that  that  these  investigators  used  a  stationary  table  and 
only  one  standard.  Having  the  weights  placed  on  a  stationary 
table  implies  that  the  so-called  Fechnerian  space  error  is 
taken  in,  whereas  it  seemed  desirable  to  avoid  this  error  in  our 
experiments.  The  use  of  only  one  standard  weight  causes  a 
rise  in  the  temperature  of  the  metal  owing  to  the  fact  that  this 
weight  is  touched  very  frequently  with  the  warm  hand.  This 
is  to  be  avoided  for  two  reasons:  First,  because  differences  of 
temperature  might  influence  in  some  way  our  judgments  of  weight, 
and  second  because  the  standard  weight  can  be  recognized  by 
being  warmer,^  whicl\  introduces  an  almost  incontrolable  in- 
fluence. The  use  of  only  one  standard  weight  may  be  less  ob- 
jectionable if  the  original  Cattell  weights  or  other  weights 
are  used  which  are  made  of  materials  which  are  bad  conductors 
of  heat.  In  experiments  in  which  the  turning  table  is  used,  it 
is  almost  indispensable  to  use  several  standard  weights. 

'The  purpose  of  our  experiments  was  in  so  far  different  from 
that  of  Martin  and  Miiller,  as  we  wanted  to  have  as  few 
disturbing  influences  as  possible  and  Martin  and  MiiUer 
were  chiefly  interested  in  the  analysis  of  the  time  and  space 
error.     It  seems  that  the  time  error  may  be  kept  constant  by 

*Martin'  und  Mueller,   7.iir     Analyse     dcr     Unterschiedsempfindliclikeit, 

1899,  p.  3  sq. 

tM.\RGARET    KeiveR    Smith,   Rythmus    und   Arbeit,    Phil.    Stud.    Vol.    IG, 

1900,  pp.  96-98. 

l.\  cisuil  observation  of  the  increase  of  the  temperature  of  the  stand- 
ard weight  by  frequent  contact  with  the  skin  was  made  by  MuELLER  and 
ScHU.M.\N.N'  Uber  die  psychologischen  Grundlagen  der  Vergleichung  gehohener 
Grufichte,  .\rch.  f.  d.  ges.  Pky.dologie,  Vol.  4.5,  18S.),  p.  112)  who,  however,  did 
not  change  their  experimental  procedure. 


14  PROBLEMS  OF  PSYCHOPHYSICS 

means  of  regulating  the  movement,  whereas  a  similar  supposi- 
tion can  not  be  made  for  the  space  error  unless  very  definite 
instructions  are  given,  which  it  is  hard  to  follow  strictly  in 
an  extended  series  of  experiments.  For  this  reason  it  seemed 
best  to  take  in  the  time  error  as  a  whole,  but  to  avoid  the  space 
error.  The  question  whether  the  time  error  may  be  eliminated 
by  an  appropriate  arrangement  of  the  computation  is  of  second- 
ary nature,  and  one  may  doul)t  whether  there  is  any  great  use 
in  doing  so.  In  this  respect  it  seems  that  our  way  of  experi- 
menting is  a  slight  improvement  over  Miiller's  method,  in 
so  far  as  it  makes  it  possible  to  avoid  the  space  error  without 
further  inconvenience  and  to  perform  the  experiments  in  com- 
paratively short  time.  It  is  not  possible  to  pass  an  equally 
favorable  judgment  on  our  modification  of  the  series  of  judg- 
ments of  which  the  subject  has  the  choice.  It  has  been  mentioned 
that  one  subject  could  not  be  trained  to  the  proper  use  of  the 
"guess "-judgments  and  it  seems  to  come  out  as  a  result  of  the 
experiments  that  the  guesses  are  more  likely  to  be  correct  than 
incorrect.  This  indicates  that  the  "g "-judgments  were  also 
for  the  other  subjects  not  equality  judgments  in  the  proper  sense 
of  the  term,  meaning  that  the  subject  had  nothing  in  his  sensa- 
tion to  go  upon  in  the  formation  of  his  judgment,  but  that  they 
were  judgments  of  a  low  degree  of  certainty.  This  is  a  fact 
which,  perhaps,  is  not  void  of  interest  but  which  has  no  direct 
bearing  On  the  problem  of  measuring  the  accuracy  of  sensations 
and,  after  having  verified  this  fact,  it  will  be  best  not  to  use  this 
distinction  between  "heavier  guess"  and  "lighter  guess".  The 
only  natural  judgment  in  these  cases  is  that  of  equality  between 
the  two  stimuli.  The  possibility  of  giving  a  "guess "-judgment 
aids  the  subject  in  his  natural  tendency  not  to  commit  himself. 
It  seems  that  a  similar  remark  must  he  made  in  regard  to 
letting  the  subject  express  the  degree  of  confidence  with  which 
the  judgments  are  given.  The  meaning  of  the  terms  "an  ordi- 
nary degree  of  certainty,"  "a  high  degree  of  certainty"  and 
"a  very  high  degree  of  certainty"  seems  to  be  clear  at  first  and 
not  requiring  any  special  definition.  In  the  actual  experimen- 
tation one  finds  very  soon  that  it  is  difficult  to  give  these  terms 
a  definite    meaning,   an  observation    which    was    also  made   by 


DESCRIPTION  OF  THE  EXPERIMENTS  15 

one  of  Miiller's  best  trained  observers.*  It  seems  that  there 
is  some  danger  of  mixing  up  two  different  problems,  the  prob- 
lem of  the  degree  of  subjective  confidence  and  that  of  the  accur- 
acy of  sensation.  It  certainly  would  be  not  only  an  interesting 
but  also  a  very  important  problem  to  find  their  relation,  but  it 
seems  that  the  factors  which  determine  the  degree  of  subjective 
certainty  are  not  yet  as  well  understood  as  they  might  be,  and 
that  other  experiments  might  be  used  more  profitably  for  set- 
tling this  problem.  The  problem  of  finding  a  measurement  of 
the  accuracy  of  sensations  is  difficult  enough  and  it  need  not  be 
made  more  complicated  by  attacking  it  from  the  most  difficult 
side.  It  seems  advisable  for  this  reason  not  to  use  other  judg- 
ments than  "heavier,"  "lighter"  and  "equal,"  unless  one  in- 
tends to  connect  the  study  of  the  exactitude  of  sensations  with 
some  other  problem.  The  degree  of  subjective  confitlence  with 
which  the  judgments  were  given  was  disregarded  entirely  in 
the  working  out  of  the  results  for  the  present  study. 

The  Tables  3-9  (Appendix  pp.  174-177)  contain  the  results 
of  the  experiments.  Each  subject  is  given  a  separate  table 
which  contains  the -judgments  for  all  the  seven  pairs  of  compar- 
ison weights  for  every  series  separately.  The  numbers  in  the 
column  T  are  the  numbers  of  "guess"-judgment3  which  are  the 
sum  of  the  "hg"-  and  the  "lg"-judgments.  The  numbers  under 
the  heading  "h"  and  "1"  give  the  number  of  "heavier"  and 
"lighter "-judgments.  The  sum  of  the  numbers  T,  h,  and  1  in 
one  line  of  a  column  is  equal  to  50.  The  different  series  are 
marked  in  the  way  explained  above.  The  numbers  at  the  bot- 
tom of  the  columns  in  the  row  marked  ^  give  the  sum  of  the 
numbers  in  a  column. 

The  numbers  of  these  tables,  as  they  stand  here,  illustrate 
plainly  the  one  fact  that  the  results  of  psychological  experiments 
show  a  great  varial^ility.  The  greatest  amount  of  care  was 
taken  to  keep  all  the  conditions  of  the  experiments  constant 
and  there  arise,  nevertheless,  variations  of  very  considerable 
size.  This  indicates  on  how  terribly  uncertain  a  basis  most  of 
the  results  rest  which  are  gained  from  psychological  experiments. 

♦Martin  und  Mueller,  1.  c  p.  9. 


16  PROBLEMS  OF  PSYCHOPHYSICS 

Pul:>lications  are  not  scarce  which  cUiira  to  demonstrate  the  in- 
fluence of  a  certain  factor  on  a  psychical  phenomenon  by  observed 
differences  in  rehitive  frequencies  amounting  to  4  or  5%.  It 
is  very  obvious  to  raise  the  objection  to  these  investigations 
that  a  similar  or  perhaps  even  a  greater  variation  of  the  results 
might  have  been  obtained,  if  the  experiments  had  been  repeated 
without  varying  the  condition  the  influence  of  which  was  to  be 
demonstrated. 

The  next  observation  is  that  there  does  not  exist  apparently 
a  difference  of  the  intensities  which  is  always  jvidged  in  the  same 
way.  A  negative  difference  of  16  grams  is  by  far  beyond  the 
difference  which  is  called  the  threshold,  but  there  occur  some 
cases  in  all  the  subjects  where  this  difference  could  not  l^e  de- 
tected or  was  judged  the  wrong  way.  A  difference  which  is  such 
that  greater  differences  are  always  judged  in  the  same  way  is 
a  fiction,  the  fact  is  that  the  judgments  of  one  Subject  on  the 
same  difference  vary,  and  it  is  impossible  to  foresee  in  which 
judgment  a  certain  experiment  will  result.  This  is  not  a  new 
observation  as  e.  g.  Miiller  and  Cattell*  in  their  discus- 
sions of  the  notion  of  the  threshold  have'  laid  stress  on  the 
fact  that  a  difference  which  is  always  judged  in  one  way  does 
not  exist.  Lately  the  view  was  taken  by  Kobileckif  that 
"there  must  be  something  wrong  with  the  experiments"  if  this 
occurs.  This  view,  if  once  taken,  is  of  course  irrefutable,  because 
it  lays  down  as  a  criterion  of  the  correctness  of  the  experiments, 
that  they  must  conform  with  the  requirement  of  a  difference  of 
intensities  which  is  always  judged  in  the  same  way,  e.  g.  that  it  is 
alwaj's  possible  to  find  a  difference  which  is  judged  equal.  The 
only  objection  is  that  it  is  not  possible  to  fiad  another  fault  with 
the  experiments  in  question  than  that  their  results  do  not  agree 
with  one's  definition  of  a  correct  experiment,  and  that  it  does 

*There  is  a  slight  diiTerence  in  the  positions  of  Catteil  and  Miiller.      Mill 
ler  takes  it  as  a  given  fact  that  there  does  not  exist  a  difference  which  is  always 
judged  in  the  same  way,  whereas  Catteil  assumes  the  suppositions    of   the 
theory  or  errors  and  argues  that  the  assumption  of  such  a  difference  is  con- 
tradictory. 

fSTAXiSLAUS  KoBYLECKi,  Uber  die  Wahrnehnibarkcit  plotzlicher  Druck- 
anderungen,  Psychologische  Studien,  Vol.  I,  1906,  p.  293,  where  the  reference 
to  Mailer's  writings  may  he  found. 


DESCKIPTIOX  OF  THE  EXPERIMENTS  17 

not  seem  to  be  quite  justified  to  discard  experiments  which  were 
performed  with  just  the  same  care  as  others,  merely  because 
their  outcome  was  not  such  as  it  ought  to  be  according  to  a  defi- 
nition laid  down  beforehand. 

These  facts  seem  to  suggest  the  following  view.  If  the  sub- 
ject is  required  to  give  judgments  on  the  comparison  of  two 
stimuli  his  judgments  will  vary,  so  that  one  can  not  possibly 
foresee  what  the  judgment  will  be  in  any  particular  experiment. 
The  only  thing  we  know  beforehand  is  that,  if  the  experiment 
succeeds,  the  judgment  vnLl  belong  to  one  of  those  classes  of 
judgments  which  were  admitted.  This  is  the  formal  character 
of  random  events  and  we  introduce  the  notion  that  there  exists 
a  definite  probability  for  every  class  of  judgment,  that  the  com- 
parison of  certain  two  stimuli  by  a  given  subject  under  well 
defined  conditions  ^\'ill  result  in  a  judgment  of  this  class.  If 
we  admit  the  judgments  "heavier",  "lighter"  and  "equal" 
we  will  have  the  probabilities  p,  q,  r  that  the  comparison  of  a 
given  difference  will  result  in  the  corresponding  judgment.  The 
numerical  values  of  these  probabilities  may  vary  from  subject 
to  subject,  and  it  is.  a  problem  of  investigation  whether  they 
remain  constant  for  the  same  subject  and  the  same  difference, 
or   whether   they  are  subjected   to    temporal    variations.*     For 


*Wreschner  defines  "trustworthiness"  of  a  sensation  (Zuverlassigkeit) 
as  the  expectations  which  a  single  sensation  or  a  judgment  on  it  has  to  be 
confirmed  in  the  case  of  repetition  under  conditions  which  are  as  nearly  alike 
as  possible.  {Methodologische  Beitrage  zu  psychophysischen  Messungen, 
Schrijten  der  Geseilschaft  jiir  psychologische  Forschung,  1895,  p.  25  "Hier- 
unter  (unter  Zuverlassigkeit)  wollen  wir  die  Aussichten  verstehen,  welche 
eine  einmalige  Empfindung  oder  deren  Beurtheilung  hat,  in  einem  Wieder- 
holungsfalle,  der  unter  moglichst  gleichen  Versuchsbedingungen  stattfindet- 
bestattigt  zu  werden....).  The  correct  analysis  of  this  notion,  of  which 
Wreschner  makes  but  Uttle  use,  might  have  led  to  the  introduction  of  the 
notion  of  a  probabiHty  of  judgments  of  certafti  type.  W'reschner's  definition 
refers  merely  to  the  subjective  side  of  the  notion  of  a  probability,  neglecting 
the  objective  definition  which  alone  is  suitable  for  the  mathematical  treat- 
ment. For  the  explanation  of  a  mathematical  probability  as  a  measure- 
ment of  the  degree  of  certainty  see  v.  Kries,  Die  Principien  der  Wahrschein- 
lichkeiisrechnung,  18.S6,  and  STfMPF,  Vber  den  Begrifj  der  mathematischen 
W ahrscheinlichkeit,  Berichte  der  bayrischcn  Akademie,  Phil.  Kl.  1892. 


18  PROBLEMS  OF  PSYCHOPHYSICS 

positive  differences  which  are  considerable  the  value  of  the  prob- 
ability of  a  ''greater "-judgment  comes  very  near  to  the  unit 
and  the  probabilities  of  other  judgments  correspondingly  are 
very  small.  For  negative  differences  which  are  considerable 
the  probability  of  a  "smaller "-judgment  is  only  little  different 
from  the  unit  and  the  probabilities  of  the  judgments  "greater" 
and  "equal"  are  very  close  to  zero.  Data  like  those  embodied 
in  the  Tabl-es  3-9  are  observations  on  the  repeated  realization 
of  chance  events  and  they  may  be  used  as  empirical  determin- 
ations of  the  underlying  probabilities.  The  question  is,  how 
far  do  the  results  comply  with  this  notion  of  a  probability  of  a 
judgment  of  certain  type,  and  how  much  can*  be  done  with  it 
for  the  solution  of  the  problems  of  psychophysics. 


CHAPTER  II. 

ON    THE    STATISTICAL    NUMBERS    OF    RELATIVE    FREQUENCY. 

A  statistical  investigation  ma}-  be  confined  to  a  description 
of  the  conditions  and  a  statement  of  the  facts  which  were  ob- 
served. Such  a  statement  is  in  itself  valualile  and  not  void  of 
interest  because  it  contains  the  description  of  a  fact,  but  as  long 
as  this  fact  is  not  connected  with  other  facts  its  statement  is  not 
so  much  knowledge  as  the  material  for  the  future  acquisition  of 
knowledge.  On  this  ground  one  even  can  not  conclude  that  under 
similar  conditions  results  will  1)e  obtained  which  resemble  those  of 
the  first  series  of  observations.  It  is,  indeed,  out  of  question  to 
reproduce  exactly  the  same  conditions  and,  since  one  does  not 
know  anything  about  the  conditions  which  necessitate  the  re- 
sult, one  can  not  positively  say  that  only  the  observed  conditions 
are  of  importance  and  one  must  resign  the  hope  to  foretell  future 
results.  But  the  main  interest  of  all  investigations  is  to  know, 
whether  the  same,  or  at  least  similar  results  will  be  obtained  in 
a  future  repetition  of  the  observation.  Before  such  a  statement 
can  1)6  made  it  is  necessary  to  form  one's  views  about  the  causes 
which  were  at  work  to  produce  the  first  result.  This  can  be 
done  by  demonstrating  the  causal  relation  between  the  results 
of  the  observation  and  a  certain  group  of  conditions  l)y  means 
of  the  method  of  experimental  variation  of  the  conditions. 
The  application  of  this  method  is  comparatively  simple  in  those 
eases  where  the  conditions  under  which  we  want  to  okserve  a 
phenomenon  are  well  under  our  control.  This  method  becomes 
more  laborious  if  our  control  of  the  conditions  is  not  complete 
enough  to  enable  us  to  establish  causal  relations;  in  these  cases 
it  becomes  necessary  to  introduce  new  notions. 

One  forms  the  hypothesis  that  the  result  is  due  to  a  complex 
of  conditions  which  remain  constant  and  another  group  of  con- 
ditions which  are  variable.     The  variations  of  the  latter  group 


20  •  PROBLEMS  OF  PSYCHOPHYSICS 

are  supposed  to  have  random  character,  i.  e.  they  do  not  show 
a  recognizable  law  or  regularity.  The  group  of  constant  condi- 
tions is  represented  liy  those  conditions  which  one  can  vary  or 
keep  constant  at  will,  or  which  l:)y  their  nature  do  not  vary,  or 
are  supposed  not  to  vary.  The  result  of  these  constant  and 
varia])le  conditions,  then,  is  determined  in  so  far  as  it  must  have 
a  certain  general  character,  but  it  is  subjected  to  quantitative 
or  qualitative  variations.  The  constant  conditions  determine 
not  only  the  class,  to  which  an  event  which  depends  on  them 
belongs,  but  they  determine  also  the  amount  of  variation  which 
may  occur  in  individual  cases.  Degrees  of  variation  may  define 
classes  which  are  sub-classes  of  the  class.  Individuals  of  these 
sul)-classes  occur  with  different  relative  frequency,  which  is  a 
fraction,  the  denominator  of  which  gives  the  number 'of  all  individ- 
uals of  the  class,  and  the  numerator  the  numl^er  of  the  individ- 
uals of  the  sub-class.  This  number,  which  is  smaller  than  one, 
has  the  character  of  a  mathematical  prol:)ability.  Every  group 
of  conditions  which  gives  certain  numerical  probabilities  to  every 
sub-class  is  called  a  system  of  causes,  or  abbreviated  a  cause  of 
an  event  of  the  class.  Different  systems  of  causes  produce 
different  results  in  so  far  as  they  may  1,  produce  events  which 
])elong  to  entirely  different  classes,  or  2,  that  they  give  numer- 
ically different  probabilities  to  the  sub-classes  of  the  same  type 
of  events.  This  terminology  differs  from  the  common  usage 
of  the  word  cause  which  designates  a  group  of  conditions  which 
necessarily  produces  a  certain  event;  here  we  mean  by  this  word 
a  group  of  conditions  which  gives  a  certain  probability  to  the 
event  in  question.  This  group  of  conditions  may  or  may  not 
be  susceptible  of  further  analysis,  but  as  long  as  further  infor- 
mation is  not  supplied,  it  is  characterized  only  by  the  numerical 
value  of  the  probability  of  the  event.  Two  groups  of  conditions 
which  give  the  same  probability  to  the  same  event  are  not  dis- 
tinguishable on  this  basis.*  If  we  know  that  in  a  series  of  obser- 
vations on  a  certain  event    the    result   of    a    single   observation 

*This  may  be  illustrated  by  the  following  example.  It  was  found  in  an 
investigation  on  the  estimation  of  long  time  intervals  by  a  great  number  of 
u  ntrained  subjects,  that  zero  was  a  favored  numeral  occurring  at  the  last 
place  much  more  frequently  than  any  other  numeral.     An  other  investiga- 


THE    STATISTICAL  NUMUERS  OF  RELATIVE  FREQUENCY  21 

depends  on  a  system  of  causes  which  is  characterized  by  the 
numerical  value  of  the  probability  of  the  event,  we  are  not  only 
able  to  state  the  most  probable  outcome  of  the  series,  but  we 
also  can  assign  a  definite  probability  with  which  we  may  expect 
a  given  deviation  from  the  most  probable  result.  We  base  our 
expectation  that  a  repetition  of  the  series  of  experiments  will  give 
similar  results  on  the  identity  of  the  conditions  which  we  know 
and  on  the  supposition  of  the  random  character  of  the  influences 
which  we  do  not  know. 

Repetitions  of  the  same  series  of  observations,  however,  do 
not  give  exactly  the  same  result  and  the  question  arises  necessar- 
ily, whether  the  differences  between  these  results  are  due  to  chance 
variations  or  whether  they  are  due  to  a  change  in  the  underly- 
ing system  of  causes.  This  question,  as  a  matter  of  fact,  is  the 
all  important  problem  of  every  statistical  investigation  which 
applies  the  method  of  experimental  variation  of  the  conditions. 
A  certain  phenomenon,  e.  g.  a  mental  state,  is  observed  under 
certain  conditions  and  from  these  observations  statistical  num- 
bers of  relative  frequency  are  derived.  One  or  several  of  the 
conditions  which  we  have  so  far  under  control  as  to  vary  them 
at  will  may  be  susceptible  of  qualitative  or  quantitative  varia- 
tion. Keeping  all  the  other  elements  constant  we  vary  these 
conditions  and  we  want  to  know,  what  the  influence  of  this  change 
is.     This  influence  is  characterized    only  by  the  differences  of 


tion  on  the  estimation  of  short  time  intervals  by  a  small  number  of  highly 
trained  subjects  showed  a  similar  great  frequency  of  zero.  If  nothing  else 
had  been  known  than  the  percentages  expressing  the  great  frequency  of  zero, 
one  had  been  obliged  to  conclude  that  similar  conditions  were  at  work  in  both 
cases,  the  conclusion  of  identical  conditions  beii.g  made  impossible  by  difi"er- 
ences  in  the  numerical  values  of  the  percentages.  Collateral  evidence,  how- 
ever, showed  that  the  conditions  involved  were  absolutely  difi'erent.  (See 
the  author's  article  on  "Systematic  Errors  in  Time-Estimation."  Amer. 
Journ.  Psychology,  1907,  Vol.  18.)  The  collateral  evidence  in  this  case 
had  the  form  of  diflerences  in  the  percentages  of  the  other  numerals,  espec- 
ially of  5  and  this  circumstance  indicated  the  possibility  and  necessity  of 
further  psychological  analysis.  It  is  not  necessary  that  the  collateral  evi- 
dence must  have  the  character  of  statistical  numbers  of  relative  frequency; 
introspective  data  or  our  general  knowledge  of  the  processes  involved  may 
make  one  hypothesis  more  likely  than  the  others. 


22  PROBLEMS  OF   PSYCHOPH Y8ICS 

the  observed  numbers  of  relative  frequency  in  the  two  series. 
The  influence  of  the  variation  of  the  concUtion  is  demonstrated, 
if  we  can  convince  ourselves  that  the  difference  is  due  to  a  differ- 
ent numerical  value  of  the  prolDability  of  the  event  in  the  second 
series  and  not  to  chance  variations.  It  is  the  object  of  further 
consideration  to  form  a  view  on  the  nature  of  this  influence,  and 
it  will  depend  on  the  particular  kind  of  problem  one  has  to  deal 
with  which  views  will  seem  acceptable.  The  real  difficulty  of 
the  treatment  of  the  statistical  data  consists  in  the  demonstra- 
tion that  the  difference  of  results  is  due  to  a  different  system  of 
causes.  The  outcome  of  the  comparison  of  the  two  series  con- 
sists in  the  difference  of  certain  numerical  values,  and  similar 
differences  would  have  been  found  if  the  conditions  had  not  been 
changed.  Are  these  differences  then  to  be  attributed  to  chance 
variations,  or  are  they  due  to  differences  in  the  objective  condi- 
tions? 

The  answer  usually  given  is  that  the  differences  must  be  ac- 
counted for  by  a  change  of  the  objective  conditions,  if  they  are 
unequivocal  and  considerable.  By  the  term  "unequivocal" 
one  means  that  the  differences  must  always  have  the  same  sign 
i.  e.  there  must  be  uniform  decrease  or  uniform  increase,  but  not 
increase  alternating  with  decfease,  if  the  group  of  conditions 
which  we  investigate  varies  in  one  way.  One  finds  it  rather 
hard  to  comply  with  this  condition  in  actual  investigations  and 
one  must  not  be  too  strict  in  the  application  of  this  requirement, 
because  even  with  phenomena  which  are  as  constant  as  the  size- 
weight  illusion  one  finds  exceptions.  As  a  rule  one  will  be  satis- 
fied if  the  percentage  of  those  cases  which  do  not  comply  with 
the  requirement  of  an  unequivocal  variation  is  small.  A  simi- 
lar vagueness  lies  in  the  requirement  of  considerable  differences. 
It  is  a  matter  of  course  that  one  can  not  speak  of  absolute  cUffer- 
ences  but  only  of  relative  or  percentual  differences.  The  abso- 
lute differences  between  the  results  of  two  statistical  series 
must  increase  if  the  numl)er  of  observations  becomes  greater, 
even  if  we  have  to  deal  with  no  other  but  chance  variations,  as  it 
follows  from  the  theorem  of  Bernoulli.  A  difference  which  may 
be  considerable  in  one  case  will  be  inconsiderable  in  another 
case  and,  furthermore,  there  exists  no  fixed  limit  where   a  nu- 


THE    STATISTICAL  NUMBERS  OF  RELATIVE  FREQUENCY  23 

merical  difference  begins  to  be  considerable.  Tlie  notion  of  a 
considerable  difference  is  too  indefinite  to  be  serviceable  for  dis- 
tinguishing between  two  results. 

The  particular  character  of  the  notions  used  does  not  permit 
a  strict  answer,  but  only  the  statement  of  the  view  which  is  more 
likely,  because  it  has  the  greater  probability.  On  the  basis  of 
the  results  one  may  form  a  number  of  different  hypotheses  which 
may  explain  both  results  but  which  have  a  different  degree  of 
probability.  It  is  obvious  at  once  that  no  difference  between 
the  results,  no  matter  how  great  it  may  be,  can  be  a  strict  dem- 
onstration of  a  difference  in  the  underlying  systems  of  causes. 
Since  the  result  of  a  single  experiment  is  not  strictly  determined 
as  far  as  our  knowledge  goes,  it  follows  that  one  has  only  a  cer- 
tain probability  with  which  one  may  expect  the  result  to  fall 
within  given  limits,  and  it  is  always  possible  that  the  opposite 
of  an  event  which  is  only  probable  arrives,  no  matter  how  near 
to  the  unit  the  probability  may  be.  Deviations  of  every  si.^e 
are  possible,  only  the  probabilities  of  their  occurrence  are  differ- 
ent. 

The  first  requirement  for  the  prediction  of  future  results  is 
that  the  observed  numbers  of  relative  frequency  have  not  only 
the  formal  but  also  the  material  character  of  mathematical  prob- 
abilities. Such  a  number  has  the  formal  character  of  a  proba- 
bility, if  it  is  a  fraction  the  denominator  of  which  gives  the  num- 
ber of  all  the  cases  and  the  numerator  the  number  of  all 
those  cases  which  are  favorable  to  the  event.  The  ratio  of  the 
number  of  the  judgments  "heavier"  to  the  total  number  of 
judgments  given  on  a  pair  of  weights  has  the  formal  char- 
acter of  a  mathematical  probability.  The  percentage  of  these 
judgments,  and  the  ratio  of  the  number  of  "heavier"-judgments 
to  the  number  of  "lighter "-judgments  have  the  character  of  func- 
tions of  mathematical  probabilities  (lOOp  and  — -).  Exam- 
ples of  numbers  of  relative  frequency  which  have  aot  the 
formal  character  of  mathematical  probabilities,  nor  of  functions 
of  such,  are  the  ratio  of  the  number  of  births  which  take  place  in 
a  country  during  a  year  to  the  number  of  inhabitants  of  this  coun- 
try, or  the  ratio  of  those  men  of  a  country  wlio  choose  a  certain 


24  PROBLEMS  OF  PSYCHOPHYSICS 

profession  to  the  number  of  inhabitants.*  Numbers  of  this  kind 
must  not  be  treated  as  mathematical  probabilities  and  certain 
important  conclusions  cannot  be  drawn  from  them. 

Neither  the  theorem  of  Bernoulli  for  constant  probabil- 
ities, nor  the  theorem  of  Poisson  for  variable  probabilities 
can  avail  for  demonstrating  that  a  statistical  number  of  relative 
frequency  has  the  material  character  of  a  probability.  The 
theorems  of  the  calculus  of  probabilities,  indeed,  do  not  contain 
any  statement  about  actual  events,  but  they  give  only  the  for- 
mal character  of  those  events  to  which  the  attribute  of  random- 
ness belongs.  The  only  possible  w^ay  of  demonstrating  that  a 
statistical  number  of  relative  frequency  has  the  material  char- 
acter of  a  mathematical  probability  is  to  show  that  this  number 
approaches  in  a  certain  way  a  definite  limit,  if  the  number  of 
observations  is  increased  indefinitely.  Let  us  suppose  we  have 
n  series  of    observations  on    an   event  E,  each    one  comprising 

s  observations  of  w^hich  m^,  m^, mn  have  given  the  result  E. 

Then  the  first  requirement  says  that  the  fractions 

s  s    '     s 

must  be  grouped  around  a  number  p  in  a  certain  way.     It  fol- 

lows  from  the  fact  that  p  is  the  limit  of  the    numbers    — -    that 

s 

they  will  be  clustered  around  p  more  closely  than  around  any 

♦For  use  of  the  birth  rate  see  Laplace,  Th eorie analytique  de  la  probability 
art.  30,  31.  Statistical  numbers  of  the  second  type  were  used  by  Mr.  Cat- 
TELL  in  his  study  of  American  men  of  science  {A  Statistical  Study  of  Amer- 
ican Men  of  Science,  III;  Science,  December  7,  1906,  N.  S.  Vol.  XXIV,  No. 
623.)  These  numbers  have  not  the  formal  -  character  of  a  mathematical 
probabiUty  because  among  the  men  living  in  a  country  there  are  also  those 
who  have  chosen  their  career,  so  that  the  number  of  inhabitants  does  not 
represent  a  totality  of  cases  which  may  go  one  way  or  the  other.  The  choice 
of  profession  of  all  the  men  born  in  a  country  during  a  certain  period  of  time 
represents  more  nearly  a  set  of  cases  which  depend  on  chance.  It  is  not  likely 
that  such  a  number  would  have  the  material  character  of  the  probability  of 
becoming  e.  g.  a  scientist  for  a  child  born  in  this  country,  because  the  denom- 
inator of  such  a  ratio  ought  to  be  the  number  of  all  those  circumstances  which 
decide  the  choice  of  a  man's  life  work,  a  totality  of  cases  to  which  no  definite 
number  can  be  assigned. 


THE  STATISTICAL  NUMBERS  OF  RELATIVE  FREQUENCY  25 

other  value.  As  to  the  law  of  the  distribution  of  these  numbers 
there  are  two  cases  possible  which  are  fundamentally  different 
for  the  interpretation  of  the  event  E:  either  the  distribution 
follows  the  (I>(^)-law  or  it  does  not.  The  interpretation  will 
be    different  according  to   the   following  cases. 

(I.)     The  distribution  follows  the  <J>(;')-law. 

1.  The  distribution  is  such  that  the  number  of  deviations, 
the  absolute  value  of  which  is  smaller  or  equal  to  y,  y  being  any 
value, 

•«L-p\<r       (.  =  1,2,  n) 


s 


is   given   by 


n  4>  ir)  ='^J   e      dx     ,  (1) 

o 

where  n  is  the  number  of  series  of  observations  and  where  h  is 
a  constant  called  the  coefficient  of  precision  which  is  given  by 


h=J—±—  .  (2) 

\2P{1-P) 

This    is  the  case   of    a   distribution    according     to    Bernoulli's 

theorem.     This   will   be   e.   g.   the   distribution   of  observations, 

where  the  single  observation  consists  in  drawing  a  ball  from  an 

urn  which  contains  black  and  white  balls  only,  if  the  ball  is  put 

back  after  its  color  was  registered.     If  p  is  not  known  one  must 

in  ■ 
take  the  arithmetical  mean  of  the  numbers —  *  . 

s 

n 

)i  s        "^  '  (3) 

I  - 1 

and  one  obtains  for  the  corresponding  coefficient  of  precision 

h'  =  J— ^  .  (4) 

V  2a(i-a) 


26  PROBLEMS  OF  PSYCHOPHYSICS 

Statistical  events  of  this  type  are  similar  to  the  drawing  of  balls 
from  an  urn  which  contains  the  same  number  of  black  balls  and 
white  balls  throughout  the  series.  The  results  of  such  a  series 
may  be  regarded  as  the  products  of  a  constant  system  of  under- 
lying causes  represented  by  those  conditions  which  determine 
the  probability  of  the  event,  and  of  another  system  which  is  sub- 
jected to  chance  variations.  A  statistical  series  of  this  kind  is 
called  by  W.  Lexis*  a  typical  series  of  normal  dispersion.  We 
are  allowed  to  suppose  that  a  repetition  of  the  observation  will 
give  a  result  which  complies  with  the  expectations  which  we  base 
on  the  calculus  of  probabilities. 

(2.)  The  distribution  follows  the  O  ( ;-)-law  but  the  precis- 
ion required  by  formula  1  is  not  equal  to  that  given  by  formula  4. 

(a.)     We  have 


In  this  case  the  values 


\  2a  {i-a) 


— '  -J)   are  less  frequent  in  the  neighbor- 

s 

hood  of  p  than  they  were  in  (1).     The  series,  however,  is  typical, 
because  the  value  p  is  fit  to  represent  the  entire  group  of  the 

I  Til  ■  ' 

numbers    j — --'P\-     Lexis    explains    this    case     by    supposing 

I    ^         I 
that  we  have  not  to  deal  with  a  constant  probability,  but  with 

one  which  undergoes  chance  variations.      Events  of  this  type  may 

be  compared  to  drawings  from  a  set  of  urns  which  were  filled 

with  white  and  black  balls  in  the  following  way.     A  coin  is  tossed 

up  n  times  and  every  time  when  head  appears  a  white  ball  is 

put  into  the  first  urn  and  when  tail  appears  a  black  ball  is  put 

into  it.     This  process  is  continued  until  every  urn  contains  n 

balls,  and  then  from  every  urn  the  same  number  of  drawings  is 

made,  the  ball  being  replaced  after  its  color  is  registered. f     The 

*W.  Lexis,  Zur  Theorie  der  Massenersckeinungen  in  der  menschlichen  Ges- 
ellschaft,  1877,  p.  22.  Cfr.  E.  Czuber,  Die  Entwicklung  der  Wahrschein- 
lichkeiistheorie ,  Jahreshericht  der  Deutschen  Mathematikervereinigung  1898, 
Vol.  7,  p.  85,  233,  and  the  same  author,  Wahrscheinlichkeitsrechnung,  1G03, 
p.  131. 

fFor  another  example  see  E.  Czuber,  Wahrscheinlichkeitsrechnung,  1903, 
p.  131. 


THE    STATISTICAL  NUMBERS  OF  RELATIVE  FREQUENCY  27 

result  of  the  drawings  from  one  urn  depends  on  the  probability 
of  the  appearance  of  a  white  ball  due  to  the  number  of  black  and 
white  balls  which  it  contains,  and  the  total  outcome  depends 
on  the  combination  of  two  chance  events.  The  result  of  this  com- 
bination is  a  distribution  according  to  the  <J>(^)-law.*  Events 
of  this  type  are  chance  events  in  the  proper  sense  of  the  word 
and  the  result  of  one  observation  does  not  depend  on  that  of  any 
of  the  previous  observations. 


(b.)     The  differences 


s 


are  distributed  as  required  by 


formula  (1.)  but  the  constant  h  is  greater  than  h'  computed  by 
formula  (4.)  In  this  case  the  differences  are  clustered  more 
closely  around  p  than  Bernoulli's  theorem  requires.  Series  of 
this  type  are  called  typical  series  with  less  than  normal  dis- 
persion. The  single  results  are  not  independent  from  each  other 
but  there  exists  a  relation  between  them.  According  to  Lexis 
such  a  distribution  is  due  to  a  law  or  a  norm,  which  tends  to  pro- 
duce a  certain  numerical  value  of  the  relative  frequency.  Events 
of  this  kind  are  not  independent  from  each  other. 


(II.)     The  distribution  of  the  terms 


s 


does  not  follow 


the  <t>(;')-law.  We  speak  in  this  case  of  an  irregular  distri- 
bution! and  we  call  such  a  series  of  observations  a  symptomatic 
series. J  These  series  indicate  a  change  in  the  system  of  under- 
lying causes.  If  the  change  occurs  always  in  the  same  direction 
we  may  speak  of  an  evolutoric  series ;  if  phases  of  increase 
alternate  with  phases  of  decrease  we  may  speak  of  an  undulatory 
variation  and,  finally,  if  the  same  changes  occur  after  constant 
time  intervals  we  may  speak  of  periodic  changes  or  variations. 
It  follows  from  this  discussion  that  the  interpretation  of  a  sta- 

*This  is  a  special  case  of  what  Bruns  calls  the  conservation  of  the  <p^T)' 
type.  See  H.  Bruns,  Wahrscheinlichkeitsrechnung  und  Kollektivmasslehre, 
1906,  p.  136. 

fL.  V.  BoRTKEWiTscH,  Das  Gesetz  der  kleinen,  Zahlen,  1898.  E.  Czubbr 
Die  Enu'icklung  der  Wahrscheinhchkcitsitheorie,  etc.,  p.  234. 

|W.  Lexis,  Massenerscheinungen,  p.  33-  A  statement  of  this  terminol- 
ogy may  be  found  also  in  E.  Czuber,  Wahrscheinlichkeitsrechnung,  p.  322 
and  E.  Blaschke,  Vorlesungen  iiber  mathemaiische  Statistik,  1906.  p,  22. 


28  PROBLEMS  OF  PSYCHOPHYSICS 

tistical  number  of  relative  frequency  depends  to  a  large  extent 
on  the  existence  of  a  limit  p  which  the  single  observations  ap- 
proach and  on  their  distribution  around  this  limit.  The  chief 
difficulty  of  deciding  this  question  in  a  particular  case  lies  in 
obtaining  a  sufficient  number  of  data,  because  one  needs  not  only 
a  large  number  of  observations,  but  even  a  large  number  of  ex- 
tended series  of  observations;  only  in  this  case  would  it  be  possi- 
ble to  compare  the  actual  distribution  with  that  required  by 
formula  (1).  It  is  not  likely  that  there  exists  in  psychology 
an  investigation  which  is  based  on  a  number  of  experiments 
large  enough  for  this  purpose;  direct  tests  of  the  O  (^)- distri- 
bution were  made  until  now  only  in  those  cases  where  favorable 
outside  circumstances  facilitated  the  task  of  collecting  material, 
as  e.  g.  in  the  statistics  of  population.  Fortunately  one  may 
devise  methods  by  which  it  is  at  least  possible  to  distinguish 
between  the  three  cases  enumerated  under  (I).  This  was  done 
by  W.  Lexis,  L.  v.  Bortkewitsch  and  E.  Dormoy.  Their 
method  is  based  on  the  computation  of  the  coefficient  of  pre- 
cision by  different  formulae.  The  first  way  of  computing  the 
coefficient  of  precision  is  indicated  by  formula  (4)  which  is  based 
on  Bernoulli's  theorem.  Each  one  of  the  n  series  can  also  be 
looked  at  as  an  observation  on  the  same  probability  p.  Ow- 
ing to  the  discrepancies  between  the  n  results  of  these  observa- 
tions it  is  necessary  to  combine  them  by  the  method  of  least 
squares,  which  is  applicable  because  we  have  to  deal  with  ob- 
servations of  the  same  empirical  constant.  If  the  number  of 
observations  is  the  same  in  all  these  n  series,  they  are  equally 
exact  empirical  determinations  of  the  probability  p.  The  co- 
efficient of  precision  is 


h"=J ^ =  J^  (5) 

and  if  we  have  to  deal  with  a  typical  probability  of  normal  dis- 
persion, it  must  be][equal  to  the  result  of  formula  (4).  The  ra- 
tio of  these  two  values 

Q  =  J ^^ (6) 

^        \(n-i)a(i-a)  ^^ 


THE  STATISTICAL  XUMBERS  OF  RELATIVE  FREQUENCY  29 

which  is  called  the  coefficient  of  divergence,  affords  the  basis  for 
the  decision  whether  we  have  to  deal  with  normal  dispersion 
or  not.  The  computation  is  not  as  laborious  as  one  mis;ht  be- 
lieve. 

The  answer  of  the  question  whether  differences  in  the  results 
of  several  series  of  observations  are  due  to  chance  variations, 
or  whether  they  indicate  a  change  in  the  system  of  underlying 
causes  must  be  based  on  the  coefficient  of  divergence.  The  case 
(II)  will  be  easiest  to  recognize  by  the  systematic  order  of  the 
deviations.  Since  this  case  indicates  a  change  of  the  conditions 
of  the  experiment,  one  may  find  the  law  of  the  variations  of  the 
probability  by  following  up  in  detail  the  variations  of  the  num- 

1)1  ' 

bers  — -    and  thus  gain  an  insight  into  the  nature  of  this  change. 

A  similar  interpretation  may  be  given  to  the  case  (I,  2b.)     The 

grouping  of    the  numbers   — -   around   p  indicates  some  recog- 

nizable  influence,  which  is  characterized  by  a  value  of  the  coeffi- 
cient of  divergence  smaller  than  1.  There  remain,  therefore, 
only  the  cases  (I.  1.)  and  (1. 2a.)  which  are  similar  in  that  respect 
that  they  do  not  indicate  a  recognizable  law.  In  the  first  case 
we  have  to  deal  with  a  constant  group  of  phenomena  and  the 
variations  of  the  observed  results  are  due  to  the  influence  of 
chance.  The  second  case  is  in  so  far  different  from  the  first  as 
it  indicates  that  the  results  are  due  to  a  group  of  phenomena 
which  itself  is  subjected  to  chance  variations. 

Our  results  are  not  extended  enough  to  compare  directly  the 
distribution  of  the  numbers  of  relative  frequency  with  that  re- 
quired by  the  probability  integral,  and  we  must  be  satisfied  with 
the  decision  whether  the  results  show  a  normal,  subnormal  or 
overnormal  dispersion.  The  first  step  to  be  taken  for  this  pur- 
pose is  the  computation  of  the  numbers  of  relative  frequency 
from  the  numbers  of  absolute  frequency  given  in  the  Tables  3-9. 
These  numbers  for  the  "heavier"  and  for  the  "lighter"  judgments 
are  given  in  the  Tables  10-16  (Appendix  pp.  177-180),  each  subject 
being  given  a  separate  table.  The  tables  are  so  arranged  that 
the  results  for  every  one  of  the  pairs  of  comparison  weights  (84, 
88,  ....  1U8)  are  given  in  a  double  column  under  the  headings  h 


30  PROBLEMS  OF  PSYCHOPHYSICS 

and  1  ("heavier"  and  "lightei-"-judgments)  in  groups  of  50  expe- 
riments. Numbers  standing  in  one  line  refer  to  results  obtained 
in  one  such  group;  thus  we  find,  for  instance,  that  for  the  subject 
VII  the  relative  frequency  of  "lighter "-judgments  for  the  com- 
parison weight  88  was  0.020  in  the  experiments  51-100  of  series 
III.  From  the  numbers  of  relative  frequenc}''  one  may  find  the 
percentages  by  multiplying  b}^  100.  These  numbers  are  empirical 
determinations  of  the  underlying  probabilities  of  "heavier"  and 
of  "lighter "-judgments  and  one  may  state  the  limits  of  accuracy 
of  such  a  series  of  50  experiments  according  to  Bernoulli's  theorem. 
The  usual  way  of  giving  the  limits  of  accuracy  in  a  determina- 
tion by  Bernoulli's  theorem  is  to  give  the  probable  errors,  which 
is  defined  by 

0.476936^^. 

This  quantity  gives  the  limits  of  the  interval  from 

'-^^  0.476936./^  to  --0.476936./^ 

inside  of  which  the  unknown  probability  p  may  be  expected 
with  the  probability  h.  The  probable  errors  for  s=50  are  given 
in  Table  17  (Appendix  p.  181)  under  the  heacUng  P.  E.  The 
first  column  of  this  table  contains  under  the  heading  m  the  num- 
bers from  1  to  25,  and  in  the  next  column  under  the  heading 
n  their  differences  from  50.  The  prol^able  error  in  the  determin- 
ation of  the  probability  of  an  event  which  was  observed  k  times 
in  a  total  of  50  experiments  may  be  found  in  the  line  which  con- 
tains the  number  k  no  matter  whether  in  the  column  m  or  in 
the  column  n. 

These  numbers  of  relative  frequency  may  be  used  still  in  an- 
other way.  The  tables  contain  data  about  series  of  observa- 
tions each  one  consisting  of  50  experiments  in  which  an  event  E 
(giving  a  "heavier"  or  a  "lighter "-judgment)  was  observed 
k  times.  One  may  ask,  which  is  the  probability  of  this  event 
in  the  next  experiment.  This  probability  is  given  by  the  for- 
mula 

k-\-i  k-\-i 

k-\-{5o-k)-\-2~    52 


THE  STATISTICAL  NUMBERS  OF  RELATIVE  FREQUENCY  31 

These  values  are  given  for  all  the  possible  results  of  a  series  of 
50  experiments  in  Table  17  under  the  headings  p'  and  q'.  Values 
of  k  which  are  smaller  than  25  must  be  looked  up  in  the  first 
column  under  the  heading  p',  those  greater  than  25  are  given 

in  the  column  q'.  The  differences  iDetween  the  values  — 
and   —    are  not  great.     The  corresponding  differences  for  s  =  300 

are  very  small  and  for  this  reason  a  similar  table  for  s=  300  is  not 
given.  The  probable  errors  of  a  series  of  300  experiments  may- 
be found  by  the  formula  given  above;  the  maximum  of  the  prob- 
able error  is  0.01947,  which  is  attained  for  k=  150. 

The  numbers  at  the  bottom  of  the  Tables  10-16  in  the  line 
marked  "Average"  are  the  arithmetical  means  of  the  numbers 
in  the  same  columns  and  they  are,  as  such,  the  numbers  of  rela- 
tive frequency  in  the  entire  series  of  experiments.  These 
numbers  show  in  a  more  convincing  way  the  same  as  the  num- 
bers of  the  smaller  series.  The  probabilities  of  "heavier"  judg- 
ments are  very  small  for  large  negative  differences  and  when 
the  differences  decrease  these  numbers  increase  at  first  slowly, 
but  then  rapidly  and  come  very  close  to  the  unit  for  some- 
what considerable  positive  differences.  With  three  subjects 
there  are  slight  breaks  in  this  upward  movement,  all  occurring 
for  large  negative  differences.  These  breaks  are  to  be  found 
in  the  tables  for  the  subjects  I,  V  and  VII  for  the  differences  84, 
and  88;  the  amount  of  inversion  is  small  in  all  the  cases.  The 
course  of  the  probabilities  of  "lighter "-judgments  is  opposite. 
These  probabilites  set  in  with  very  high  values  for  large  nega- 
tive differences,  decreasing  slowly  at  first,  but  coming  down  rap- 
idly to  very  small  values  for  positive  differences. 

The  numbers  given  in  the  Tables  10-16  serve  for  the  compu- 
tation of  the  coefficient  of  divergence.  The  first  business  is  to 
find  the  deviations  and  the  squares  of  the  deviations  from  the 
arithmetical  mean.  After  having  found  the  sum  of  the  squares 
of  the  deviations  we  are  able  to  determine  the  physical  precision 
by  formula  5;  the  values  of  the  combinatoric  precision  may  be 
found  immediately  from  the  data  in  Tables  10-16  by  the  applica- 
tion of  formula  4.     The  values  of  h'  and  of  h"  are  given  in  the 


32  PROBLEMS  OF  PSYCHOPHYSICS 

corresponding  column>  of  the  Tables  18-24  (Appendix  pp.  181-184). 
Dividing  h'  by  h"  gives  the  value  of  the  coefficient  of  di- 
vergence which  is  given  in  the  tables  in  the  columns  Q.  A 
glance  at  these  numbers  shows  that  they  vary  not  inconsiderably, 
and  that  the  range  and  the  amount  of  variation  is  very  different 
for  the  different  subjects.  The  averages  of  the  coefficients  of 
divergence  are  given  here  for  the  different  subjects: 

Subject 


II 

1.01 

VII 

1.13 

I 

1.22 

VI 

1.39 

IV 

1.50 

V 

1.62 

III 

1.69 

These  numbers  seem  to  indicate  that  we  have  to  deal  in  the  cases 
of  subjects  IV,  V  and  III  with  a  slightly  overnormal  dispersion.  It 
may  be  doubtful  whether  subjects  I  and  VI,  are  cases  of  normal  or 
slightly  overnormal  dispersion,  but  for  the  subjects  II  and  VII  it 
is  almost  certain  that  their  result  show  a  normal  dispersion.  For 
subject  II,  for  instance,  7  of  the  Q's  are  above  and  7  are  below 
the  unit  and  if  one  disregards  the  result  for  10 4h,  which  is  clearly 
out  of  bounds,  the  results  vary  between  0.76  and  1.43.  With 
this  one  may  compare  the  numbers  for  the  coefficients  of  diver- 
gence for  the  rate  of  male  and  female  births,  which  were  found 
by  Lexis*  in  his  first  series  of  observations.  They  are  based 
on  the  statistics  of  the  birth  rates  in  34  Prussian  counties  during 
24  months.  This  gives  34  series  of  24  observations  each;  one 
series  consists  of  from  639  to  4,766  observations.  The  values  of 
the  Q's  vary  between  0.72  and  1.47  and  19  of  them  are  beyond 
and  15  below  the  unit.  The  range  of  variation  is  therefore  a 
little  larger  than  in  our  experiments,  if  one  omits  the  result  for 
104h,  and  our  results  compare  very  favorably  even  if  one  does  not 

*W.  Lexis,  Das  Geschlechtsvcrhaltiiiss  der  Geborenen  und  die  Wahrschein- 
liohkeitsrechnung,  pp.  216-245;  Zur  Theorie  der  Massenerscheinungen,  pp  64.78; 
the  results  are  reproduced  in  E.  Czuber,  Wahrscheinlichkeitsrechnung  und 
ihre  Anivcndung,  1903,  p   324. 


THE  STATISTICAL  NUMBERS  OF  RELATIVE  FREQUENCY  33 

omit  it.  The  arithmetical  mean  of  the  Q's  in  Lexis's  observa- 
tions is  1.09,  approaching  more  closely  the  less  satisfactory 
result  for  subject  VII  than  that  for  subject  II.  The  slight  devi- 
ations from  the  unit  may  be  well  accounted  for  by  the  inaccuracy 
in  the  determination  of  h".  Considering  the  fact  that  Lexis's 
results  are  based  on  24  series  of  observations  each  one  contain- 
ing at  least  13  times  as  many  experiments  than  ours  (the  aver- 
age of  the  number  of  "experiments"  in  one  series  of  Lexis  is 
approximately  2,300  or  46  times  as  many  than  in  our  experiments) 
one  will  be  justified  in  believing  that  the  results  for  the  subjects 
II,  VII,  I  and  perhaps  also  those  for  subject  VI  indicate  a 
normal  dispersion.  The  dispersion  in  the  results, for  the  other 
subjects  may  be  considered  to  be  overnormal.  We  formulate 
this  result  in  this  way:  The  statistical  numbers  of  relative 
frequency  which  one  obtains  in  psychological  experiments  show 
for  some  subjects  a  normal  dispersion.  According  to  our  pre- 
vious discussion  we  will  conclude  that  the  group  of  conditions 
on  which  the  formation  of  a  judgment  depends  may  remain 
fairly  constant  for  certain  subjects.  The  answer  of  the  question 
whether  or  not  we  have  to  deal  with  a  typical  probability  cannot 
be  undertaken  on  the  basis  of  our  present  material;  it  must  be 
postponed  until  further  results  are  at  hand. 

The  demonstration  of  the  normal  dispersion  of  the  results  is 
interesting  in  several  respects.  At  first,  the  number  of  examples 
in  which  a  normal  dispersion  could  be  demonstrated  is  not  great. 
Besides  this  pretium  raritatis  there  comes  in  the  practical  conclu- 
sion that  experimental  psychology  is  certainly  capable  of  be- 
coming as  much  of  a  science  as  any  other  branch  of  knowledge 
which  takes  its  material  from  statistical  results.  The  fact  that 
psychology  has  to  deal  with  probabilities  of  normal  dispersion 
gives  this  science  a  distinguishing  feature  of  which  not  many 
statistical  sciences  can  boast  at  present.  The  normal  dispersion 
of  the  numbers  of  relative  frequency  also  proves  that  the  instabil- 
ity of  mental  states  is  not  so  very  great  as  some  writers  were  in- 
clined to  believe.  For  some  time  the  view  was  very  popular  among 
certain  philosophers,  that  a  mental  state  may  be  connected  with 
very  widely  different  physical  conditions,  and  that  the  same 
groups  of  physical  conditions  may  be  connected  with  different 


34  PROBLEMS  OF  PSYCHOPHYSICS 

mental  states.  Our  results  show  that  at  least  the  latter  part 
of  this  statement  is  not  quite  correct,  for  in  some  subjects  the  re- 
sults are  such  as  they  would  be,  if  they  were  chance  phenomena 
under  the  influence  of  a  constant  group  of  conditions.  Training 
in  psychological  experiments  seems  to  have  the  tendency  of  produc- 
ing in  a  sul^ject  the  ability  of  reacting  under  certain  influences,  and 
we  may  define  a  subject  with  training  in  psychological  experiments 
as  a  subject,  which  is  capable  of  reacting  in  such  a  way  that  the 
reactions  are  due  to  a  constant  group  of  conditions  w^hich  are 
so  far  under  the  control  of  the  subject  that  they  may  be  varied 
at  the  will  of  the  experimenter.  Subject  II  who  has  the  smallest 
coefficient  of  divergence,  had  several  years  experience  in  psycho- 
logical experiments.  Subjects  VII  and  I  had  considerable  abil- 
ity for  this  work,  whereas  subject  III  was  the  man  who  could  not 
be  trained  to  the  proper  use  of  the  "guess "-judgments,  and  sub- 
ject V  was  a  young  lady  with  little  practice  in  experimental  work. 
The  observed  values  of  the  coefficients  of  divergence  prove 
that  there  is  no  appreciable  change  in  the  results  which  are  taken 
at  different  times  for  at  least  some  subjects.  The  next  question 
is  whether  there  do  not  occur  any  changes  within  a  series.  Be- 
sides the  variability  of  the  conditions  of  a  single  lifting  (e.  g. 
variations  in  the  velocity  or  height  of  lifting)  there  is  one  circum- 
stance which  might  be  of  influence  on  the  results:  Namely 
the  taking  of  the  experiments  in  groups  of  35.  It  takes  about 
three  minutes  to  perform  such  a  series  and  one  may  ask  whether 
this  does  not  produce  an  influence  (fatigue,  strain  of  attention,) 
which  puts  the  judgments  in  the  latter  part  of  one  series  under 
conditions  which  are  entirely  or  partially  different  from  those 
in  the  preceding  part  of  the  series.  One  may  try  to  settle  this 
question  by  means  of  the  introspective  evidence  that  there  was 
or  that  there  was  not  an  appreciable  influence  of  fatigue  or  lack 
of  attention.  It  is  very  likely  that  this  question  refers  to  con- 
ditions which  are  not  favorable  to  introspection,  and  we  take 
the  attitude  that  an  objection  raised  against  a  series  of  experi- 
ments on  the  ground  of  introspective  evidence  makes  the  series 
suspect,  but  that  the  absence  of  any  such  objection  based  on 
introspection  does  not  put  the  result  beyond  doubt.  The 
introspective  evidence,  as  a  matter  of  fact,  forms  only  one  part 


THE    STATISTICAL  NUMBERS  OF  RELATIVE  FREQUENCY  35 

of  the  outcome  of  the  experiments.  The  objective  records  are  the 
other  part,  and  it  would  be  one-sided  to  neglect  one  part  for 
the  sake  of  the  other.  This  attitude  is  similar  to  that  which  one 
takes  towards  sets  of  measurements  which  are  the  results  of 
observations  of  a  physical  quantity,  where  an  opinion  about 
the  value  of  the  observations  formed  in  collecting  the  data  must 
be  verified  by  a  minute  examination  of  the  results  in  order  to 
establish  the  objective  value  of  the  .results.  There  were  in  our 
experiments  no  spontaneous  complaints  from  the  part  of  the 
subjects  as  to  the  length  of  the  experiments  and  cautious  ques- 
tioning never  revealed  any  discomfort  caused  by  the  protracted 
experimentation.  Introspective  evidence  against  our  experiments 
is  therefore  not  at  hand. 

A  change  in  the  conditions,  which  is  not  noticed  by  introspec- 
tion, must  be  expressed  by  a  variation  of  the  results  i.e. in  our  case 
by  a  different  frequency  of  the  "heavier",  "lighter"  and  "guess"- 
judgments  in  the  different  parts  of  one  series  of  35  experiments. 
We  shall  make  this  investigation  on  the  numbers  for  the  combined 
h  and  hg  judgments.  It  happens  that  the  records  of  the  single 
series  are  naturally  divided  in  five  groups,  each  one  containing 
the  judgments  given  during  one  turn  of  the  table,  i.  e.  one  judg- 
ment for  every  pair  of  comparison  weights.  The  number  of 
"heavier "-judgments  given  during  one  turn  of  the  table  can  be 
found  by  counting  these  judgments  in  one  column  of  the  records. 
Dividing  the  number  of  "heavier "-judgments  in  the  1,2,  ....5  col- 
umn by  the  total  number  of  judgments  of  a  column  one  finds  the 
relative  frequency  of  these  judgments  during  the  first,  second. 
....fifth  part  of  a  series.  These  five  numbers  of  relative  fre- 
quency are,  for  one  individual,  only  slightly  different,  as  may  be 
seen  from  Table  25  (Appendix  p.  185).  The  numerals  at  the  head 
of  the  columns  (1-5)  refer  to  the  five  turns  of  the  table  in  one  series 
and  the  numbers  in  one  column  give  the  relative  frequency  of  the 
"heavier "-judgments  during  the  first,  second,.. ..fifth  turn  of  the 
table  for  the  different  subjects.  The  variations  of  the  numbers 
referring  to  one  person  are  small  and  they  are  not  systematic, 
increase  and  decrease  interchanging  without  regularity. 

The  total  number  of  judgments  given  during  one  particular 
turn  of  the  table  was  for  the  subjects  I,  II  and  III,  840  and  for 


36  PROBLEMS  OF  PSYCHOPHYSICS 

the  other  subjects  560.     The  numbers  in  one  line  of   Table  25 

give  the  ratios  —  {i=  1,  2,  3,  4,  5),  where  ni;  is   the  number  of 
s 

"heavier "-judgments   in   the  columu  i.     Let  us  call  the  arith- 
metical mean  of  these  numbers  (referring  to  one  subject)  p  a:id 

designate  by  ^( — '-/>)     the  sum  of  the   squares  of  the  d3vi- 

f 
in  ■ 
ations  of  the  numbers  — '  from  their  mean.     We  define  further- 

<. 

more,  by  q  the  probability  of  a  judgment  which  is  not  a  "heavier" 

judgment   so   that    q=  1-p,  and  we  designate  by  u  the     mean 

error  of  the  arithmetical  mean  of  the  numbers  — * 


-ViPe^ 


These  numbers  may  be  found  from  the  data  of  the  preceding 
table  and  are  given  under  the  corresponding  headings  of  Table  27. 
They  serve  for  the  determination  of  the  coefficient  of  divergence 
Q,  which  in  problems  of  this  kind  is  found  by  the  formula 


6.^/.+  -^'-^™'-^).  (8) 


We  determine,  furthermore,  a  quantity  M  which  is  defined  by 


M 


-Vfi-^e-^)-  «) 


This   quantity  is   called  the    component   of  physical    variation* 
because  it  indicates  the  amount  of  variations   of  the  underlying 

♦This  term  (Physische  Schwankungskomponente)  is  due  to  W.  Lexis, 
Ubcr  die  Theorie  der  Stabilitat  statistischer  Reihen,  Jahrbtich  ^  .National  Okon- 
omie  u.  Statistik,  Vol.  32,  1879.  L.  v.  BorTkewiTsch,  Das  Gcstez  der  kleinen 
Zahlen,  1898,  p.  30,  calls  the  quantity  M  "absoluter  Fehlerexcedent. "  Cfr. 
E.  CzuBER,  Wahrscheinlichkeitsrechnung,  1903,  pp.  317-321. 


THE   STATISTICAL  NUMHERS  OF  RELATIVE  FREQUENCY 


37 


complex  of  causes  which  determines  the  probability  of  the  event 
in  question  in  the  different  parts  of  the  series.  M  and  Q  are  in  a 
simple  relation  shown  by  the  formula 


where  the  factor  u'  represents  the  quantity 

These  relations  may  be  used  for  checking  the  computation  by 
the  first  formula  for  M. 

We  will  illustrate  the  course  of  the  computation  by  working  out 
the  results  for  the  first  subject.  The  total  number  of  judgments 
in  each  one  of  the  five  columns  was  840,  and  the  numbers  of 
"heavier"  and  "heavier  guess"  judgments  are  given  in  the 
following  little  table  under  the  heading  m^.  By  dividing  these 
numbers  by  840  one  obtains  the  relative  frequencies  of  these 
judgments  in  the  five  columns;  the  arithmetical  mean  of  these 
numbers  is  found  to  be  p  =  0.5374.     The  column  under  the  head- 

ing   — *- — h   gives  the  deviations  from  the  arithmetical  mean. 
^    840    ^   ^ 

The  sum  of  the  positive  deviations  is  0.0449  and  the  sum  of 
the  negative  deviations  is  0.0450;  this  difference  is  due  to  the 
abbreviation  of  the  result.  The  last  column  of  the  table  gives 
the  squares  of  the  deviations,  the  sum  of  which  is  found  to  be 
0.00202773. 


472 
461 
444 
459 
424 


840 

0.5619 
0.5488 
0.5250 
0.5464 
0.5048 


840 


-^b-P 


\  840 


P)' 


+  0.0245 

0.00060025 

+  0.0114 

0.00012996 

-0.0124 

0.00015376 

+  0.0090 

0.00008100 

-0.0326 

0.00106276 

m- 


840 


2.6869 


^/j^_  \.  ^  0.00202773 


840 


p=  0.5374   I  \l{'^^-py=^- 
5    ^8ao  ^/ 


00040555 


840 


38  PROBLEMS  OF  PSYCHOPHYSICS 

Dividing  the  sum  of  the  squares  of  the  deviations  by  5.4=20 
and  taking  the  square  root  one  finds  the  number  w  =  /y/ 0.000 1039 
=  0.0101,  the  mean  error  of  the  arithmetical  mean.  Now  one 
proceeds  to  find  M  by  means  of  formula  9. 

log  839=2.92376 
log  840=2.92428 

Difference        =-0.00052 
log   0.00040555=  0.60804-4 

Sum  (regarding 
•     the  sign)  =  0.60752-4 


Dividing  by  2        =  0.30376-2  =  log  M 
M=  0.020126 

The  computation  of  the  quantity  Q  by  formula  (8.)  and  the  check 
of  the  computation  of  M  may  be  effected  in  this  way. 

log   0.00040555=0.60804-4  log  0.5374  =  0.73030-1 

log   839  =2.92376  log  0.4626  =  0.66521-1 


Sum                =0.53180-1 

Sum    =0.39551-1  = 

log  pq             =0.39551-1 

(Q^- 

-1) 

=  log  pq 

Difference      =  0. 13629=  log 

Q=V  2-1369  =  1.462 

log  (Q^-l)  =  0.13629 
log  \/Q'-l=  0-06814 

log  pq     =  0.39551-1 

log  840    =2.92428 

Difference     =0.47123-4 

Dividing  by  2     =0.23562-2 
logv'Q'-l        =0.06814 

Sum  =  0.30376-2=  log  M,  as  above. 


THE  STATISTICAL  NUMBERS  OF  RELATIVE  FREQUENCY  39 

In  Table  26  (Appendix  p.  185)  data  are  given  whicli  facilitate 
the  control  of  the  computation;  the  denotations  used  in  this 
table  are  the  same  as  those  used  in  the  text.  The  final  results 
for  the  coefficients  of  divergence  and  for  the  components  of  phys- 
ical variation  are  given  in  Table  27,  (Appendix  p.  185)  under 
the  corresponding  headings.  It  must  be  kept  in  mind  that  in 
problems  like  this,  where  the  coefficient  of  divergence  is  computed 
by  formula  (8),  the  result  must  necessarily  be  a  value  greater 
than  the  unit,  and  one  can  require  only  that  it  does  not  exceed 
this  value  too  much.  The  value  Q=\/2=  1-414  is  the  limit 
which  ought  not  to  be  exceeded,  because  in  this  case  the 
physical  variation  is  equal  to  the  accidental  variation.  Only 
in  two  of  our  subjects  do  the  Q's  exceed  this  limit  (I  and  VI) 
the  margin  being  small  (0.048  and  0.164.)  The  average  of  all 
the  Q's  is  1.345  which  comes  very  near  the  case  of  equality  of 
physical  and  accidental  variation  the  difference  being  on  the 
safe  side.  It  will  be  noticed  that  for  the  subjects  II  and  VII, 
for  whom  we  have  found  above  a  normal  dispersion,  the  values 
of  Q  come  nearest  to  1.414,  the  differences  being  0.006  and  0.018. 
From  this  it  follows  that  there  is  no  difference  between  the  re- 
sults of  experiments  made  at  different  parts  of  a  series  of  35. 


CHAPTER  III. 

ON    THE    METHOD    OF    JUST    PERCEPTIBLE    DIFFERENCES.* 

The  method  of  just  perceptible  differences  is  the  oldest  of  all 
psychophysical  methods.  After  attention  was  called  to  the 
fact  that  our  judgments  on  the  equality  of  two  stimuli  are  not 
exact  in  any  strict  sense  of  the  word,  it  was  obviously  possible 
to  investigate  the  limits  within  which  two  stimuli  may  vary  with- 
out a  difference  being  perceived.  This  was  indeed  the  problem 
of  Lambert t  and  BouguerJ,  the  first  investigators  who  tried 
to  determine  this  range  of  objective  variability  with  subjective 
equality  for  optical  sensations.  When  Lambert  lighted  a  screen 
by  a  candle  placed  in  front  of  the  centre  and  tried  to  find  the 
range  inside  of  which  there  was  no  appreciable  difference  in  the 
intensity  of  the  light,  he  had  in  his  result  a  quantity  essentially 
identical  with  the  threshold  of  difference  as  determined  by  the 
method  of  just  perceptible  differences.  This  method  came  more 
into  prominence  by  the  experiments  of  Weber  and,  later  on,  of 
Fechner,  who  used  the  notion  of  a  just  perceptible  difference 
for  his  theory  of  the  measurement  of  sensations.  The  notion 
of  the  threshold  became  identical  with  that  of  the  just  perceptible 
difference,  which  was  defined  as  the  smallest  difference  which 
could  be  perceived.  The  fact  that  smaller  differences  may  be  per- 
ceived and  that  larger  differences  may  not  be  detected  was  ex- 
plained by  accidental  errors.  The  notion  of  the  just  perceptible 
difference  thus  became  the  fundament  of  quantitative  psychology 
and  this  method  is  used  directly  or  indirectly  in  most  of  the 
investigations  which  deal  with  Weber's  Law  or  with  the  problem 

*A  short  abstract  of  this  chapter  was  given  in  the  Psychological  Review, 
Vol.  14,  July,  1907,  pp.  244-253. 

ILambert,  Photomeiria,  sive  de  mcnsura  et  gradibus  luminis,  colorum  et 
umbrae,  1760,  in  OsTWALD's  Klassiker  d.  ex.  Naiurwiss,  31-33. 

iBouGUER,  Traits  d'optique  sur  la  gradation  de  la  lumikre,  \7C<0. 

40 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  41 

of  the  accuracy  of  sense  perception,  in  spite  of  the  fact  that  there 
is  at  present  a  tendency  to  substitute  one  of  the  error  methods 
for  the  method  of  just  perceptible  differences  in  accurate  work. 
The  form  in  which  the  method  of  just  perceptible  differences 
was  used  by  the  first  investigators  was  open  to  many  criticisms 
and  it  lasted  some  time  until  this  method  obtained  the  form  in 
which  it  is  used  to-day.  The  merit  of  this  perfection  is  chiefly 
due  to  Fechner,  who  also  gave  the  name  to  this  method.  In 
determining  the  threshold  by  the  method  of  just  perceptible 
differences*  one  starts  from  two  stimuli  which  are  equal  and, 
keeping  one  stimulus  constant,  one  increases  the  other  until  a 
difference  is  perceived.  This  difference  is  put  down  as  the  result 
of  an  observation  on  the  just  perceptible  positive  difference. 
The  second  step  consists  in  finding  the  just  imperceptible  posi- 
tive difference.  This  is  done  by  allowing  a  supraliminal  differ- 
ence to  decrease  until  it  ceases  to  be  perceptible.  The  largest 
difference  which  is  not  noticed  is  put  down  as  an  observation 
of  the  just  imperceptible  positive  difference.  The  just  percep- 
tible and  the  just  imperceptible  positive  differences  are,  gener- 
ally speaking,  not  identical.  In  a  similar  way  one  finds  the  just 
perceptible  and  the  just  imperceptible  negative  differences.  In 
an  investigation  which  aims  at  some  accuracy  it  is  indispens- 
able to  make  a  considerable  number  of  experiments  in  order  to 
eliminate  the  errors  of  observation  which  express  themselves  in 
differences  of  the  numerical  results.  It  seems  to  l)e'  justified  to 
treat  the  results  by  the  algorithm  of  the  method  of  least  squares,. 
since  one  considers  the  just  perceptible  difference  a  physical 
quantity  the  observations  of  which  are  subjected  to  chance 
errors.  The  probable  error  of  the  final  result  may  serve  as  an 
indication  of  the  greater  or  smaller  variability  of  the  psycho- 
physical conditions.     The  arithmetical  mean  of   the  just  percep- 

*For  the  description  of  this  method  see:  Fechxer,  Uber  die  psychischen 
Massprincipien,  Phil.  Stud.  Vol.  IV,  1888,  p.  161  sqs.  besides  the  "Eletnente" 
Vol.  I  and  II  and  the  "Revision" ;  Wundt,  Das  ]]'ehersche  Gesetz  uttd  die 
Methode  der  Minimaldnderungen,  Phil.  Stud.  Vol.  I,  pp.  556  sqs.  Physiolo- 
gische  Psychologic,  5  ed.,  \'ol.  I,  pp.  470  sq.,  475-479;  TitchenER,  Experi- 
mental Psychology,  1905,  Vol.  II,  Part  I,  pp.  55-69,  and  Part  2,  pp.  99-143 
with    historical   notes. 


42  PROBLEMS  OF  PSYCHOPHYSICS 

tible  positive  and  of  the  just  imperceptible  positive  difference 
defines  the  threshold  in  the  direction  of  increase,  and  the 
arithmetical  mean  of  the  jusc  perceptible  and  of  the  just  im- 
perceptible negative  difference  gives  the  threshold  in  the  direc- 
tion of  decrease.* 

The  practical  application  of  this  method  meets  with  two  very 
peculiar  difficulties  which  impair  its  serviceability  and  necessi- 
tate a  change  in  the  experimental  procedure.  It  seems  that 
it  is  essential  for  the  result,  by  which  steps  the  threshold  is  ap- 
proached. After  the  first  rough  determination  is  made,  one  tries 
to  get  a  more  accurate  one  by  using  smaller  intermediate  steps. 
In  experimenting  with  this  new  series  one  notices  very  soon 
that  the  subject  is  more  apt  to  perceive  smaller  differences  in 
the  determination  of  the  just  perceptible  dif^'erenc^,  and  not  to 
perceive  larger  differences  in  the  determination  of  the  just  im- 
perceptible difference  than  he  was  before.  Usually  one  explains 
this  fact  by  the  influence  of  expectation,  because  the  subject 
knows  that  an  imperceptible  difference  is  to  be  increased  and 
a  perceptible  difference  is  to  be  diminished.  In  order  to  avoid 
this  inconvenience  one  has  employed  two  means.  The  first  was 
to  use  only  trained  subjects  who  are  free  from  the  influence  of 
expectation,  and  the  second  consisted  in  applying  the  stimuli 
in  irregular  order.  The  first  way  cannot  always  be  used,  because 
one  frequently  is  obliged  to  experiment  on  untrained  ot^alf- 
trained  subjects,  and  this  requirement  of  training  on  tlte  part 
of  the  subject,  furthermore,  disposes  of  the  possibility  of  using 
this  method  for  practical  purposes.  The  determination  whether 
the  sensitivity  of  a  patient  is  below  or  above  the  normal  must 
necessarily  be  made  on  untrained  subjects,  and  until  now  no 
other  method  but  the  method  of  just  perceptible  differences 
could  be  used  for  this  purpose.  The  method  of  presenting  the 
comparison  stimuli  in  irregular  order  has  the  inconvenience  that 
the  results  cannot  be  worked  out  by  the  algorithm  of  the  method 

*The  threshold  m  the  direction  of  increase  and  that  in  the  direction  of 
decrease  are  frequently  combined  in  order  to  obtain  a  generalized  threshold 
or  to  eliminate  constant  errors.  This  procedure,  which  was  first  suggested 
by  VoLKMANN,  gi\es  a  result  which  seemed  to  be  not  entirely  clear  in  its 
signification;  see  Titchner,  /.  c.  Part  2,  p.  112  sqs. 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  43 

of  just  perceptible  differences.  This  modification  of  the  method 
of  just  perceptible  differences  is  called  the  method  of  irregular 
variation.  The  results  obtained  in  this  way  are  usually  worked 
out  by  an  algorithm  similar  to  that  of  the  error  methods.* 

The  second  difficulty  is  due  to  the  fact  that  frequently  a  differ- 
ence is  not  noticed  after  a  smaller  one  has  been  perceived  in  a 
series  for  the  determination  of  the  just  perceptible  difference, 
or  that  a  difference  is  noticed  after  a  larger  difference  has  been 
imperceptible  in  a  determination  of  the  just  imperceptible  differ- 
ence.! The  question  arises,  what  must  be  done  with  such 
results?  Some  investigators  take  the  view  that  these  series 
must  be  ruled  out  and  that  only  those  series  must  be  kept 
for  the  final  computation,  where  no  such  inversion  occurs 
and  where  the  two  classes  of  judgments  are  strictly  sep- 
arated. Another  possibility  consists  in  not  going  beyond  the 
first  difference  which  is  perceived  or  which  fails  to  be  perceived, 
thus  avoiding  these  dubious  cases.  It  is  obvious  of  course  that 
one  may  try  to  escape  this  difficulty  by  taking  very  large  inter- 
mediate steps,  but  usually  one  is  afraid  of  doing  this,  because  one 
seems  to  renounce  the  hope  of  obtaining  a  determination  of  the 
threshold  the  accuracy  of  which  may  compare  favorably  with 
that  of  determinations  by  one  of  the  error  methods. 

To  these  practical  difficulties  comes  the  theoretical  problem 
of  finding  the  relation  between  the  results  of  the  method  of  just 
perceptible  differences  and  those  of  the  error  methods,  especially 
of  the  method  of  right  and  wrong  cases.  This  question  is  a  very 
urgent  one  since  the  individual  experiments  are  the  same  for 

*\VuNDT,  Physiologische  Psychologic,  5  ed.,  1902,  Vol.  I,  p.  478.  Some 
criticism  of  Wundt's  view  on  the  relation  of  the  method  of  just  perceptible 
differences  to  the  method  of  irregular  variation  may  be  found  in  E.  B.  Holt, 
Classification  of  Psychophysical  Methods,  Psych.  Review,  Vol.  XI,  Nov.  1904, 
p.  348,  who  contends  that  the  method  of  irregular  variation  does  not  ap- 
proach the  error  methods,  as  Wundt  says,  but  that  it  is  identical  with  them. 
He  furthermore  remarks  that  the  method  of  just  perceptible  differences  is 
not  a  method,  if  this  word  is  used  in  its  proper  sense,  but  the  statement  of 
the  intention  to  find  a  significant  value  for  the  threshold. 

tSeveral  examples  of  such  series  were  published  lately  by  Wilhelm 
Specht,  Da^  Verhalten  von  U titer schiedsschwelle  und  Reizschuelle  im  Gebiete 
des  Gehorsinnes,  Archiv.  j.  d.  ges.  Psychologic,  Vol.  9,  1907,  p.  207. 


44  PROBLEMS  OF  PSYCHOPHYSICS 

both  methods.  The  difference  merely  consists  in  using  several 
pairs  of  comparison  stimuli  in  a  particular  order  in  the  method 
of  just  perceptible  differences,  or  in  any  order  in  the  method  of 
irregular  variation,  and  only  one  pair  in  the  method  of  right  and 
wrong  cases.  To  this  comes  that  the  results  of  the  error  methods, 
as  well  as  those  of  the  method  of  just  perceptible  differences 
seem  to  confirm  Weber's  Law,  in  spite  the  fact  that  they  are 
not  comparable  among  each  other.  Some  investigators  believed 
that  the  results  of  the  method  of  right  and  wrong  cases  were 
in  no  direct  relation  to  those  of  the  method  of  just  perceptible 
differences,  but  others  tried  to  establish  such  a  relation  by  math- 
ematical formulae;  this  relation  however  proved  to  be  of  very 
complicated  nature.  Of  course  one  might  take  the  view  that 
one  of  these  methods  is  not  legitimate,  but  this  view,  which  is 
somewhat  narrow,  would  be  justifiable  only  if  the  interpretation 
of  the  results  of  either  one  of  these  methods  were  absolutely 
clear.  The  difficulties  of  the  method  of  just  perceptible  differ- 
ences, which  were  mentioned  above,  are  considered  strong  argu- 
ments against  the  use  of  this  method  and  at  present  many,  if 
not  most  psychologists  would  favor  the  error  methods  against 
the  method  of  just  perceptible  differences,  if  they  had  to  choose 
between  them,  in  spite  the  fact  that  the  foundations  of  the  method 
of  right  and  wrong  cases  are  little  known  and  not  very  well  under- 
stood. 

For  the  development  of  the  psychophysical  methods  one 
circumstance  proved  to  be  of  fundamental  importance:  The 
introduction  of  the  theory  of  errors  by  Mbbius  and  Fechner, 
which  necessitates  the  division  of  the  judgments  into  the  two 
classes  of  correct  and  wrong  cases.  The  theory  of  errors  of 
observation  gave  some  insight  into  the  nature  of  the  error  meth- 
ods and  one  could  hope  to  find  the  relation  of  these  methods  to 
the  method  of  just  perceptible  differences,  because  the  Gaussian 
coefficient  of  precision  (the  "mensura  j.recisionis")  seemed  to 
give  a  measure  of  the  accuracy  of  sensations  similar  to  that 
afforded  by  the  smallest  perceptible  difference.  It  escaped 
attention  for  a  long  time  that  the  application  of  the  theory  of 
errors  of  observation,  though  helpful  for  certain  purposes,  en- 
tirely excludes  the  notion  of  a  just  perceptible  difference.     It 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  45 

is  the  merit  of  Jastrow  and  Cattell  to  liave  called  attention  to 
this  fact.  The  theory,  indeed,  starts  from  the  supposition  that 
the  probability  of  every  error  is  a  function  of  the  si^e  of  this  er- 
ror; the  theory  makes  certain  assumptions  as  to  the  nature  of 
this  dependence,  which  imply  that  no  error,  no  matter  how  large, 
is  impossible  although  its  probability  may  be  very  small.  One 
also  finds  that  the  greatest  error  which  is  likely  to  be  committed 
in  a  certain  series  of  observations  depends  on  the  number  of 
observations,  so  that  the  more  extended  the  series  is  the  greater 
the  largest  error  becomes  which  is  likely  to  be  committed.  Cat- 
tell,  supposing  that  the  errors  in  judgments  on  differences  of 
intensity  of  two  stimuli  follow  the  same  law  as  the  errors  of  ob- 
servation,* concludes  that  there  does  not  exist  a  just  percep- 
tible difference  in  any  absolute  sense  of  the  term,  because  the 
smallest  difference  beyond  which  all  the  judgments  of  a  series  are 
correct  depends  on  the  number  of  observations  in  this  series. 
The  supposition  that  there  exists  a  difference  which  is  always 
perceived  is,  therefore,  in  contradiction  with  the  fundamental 
supposition  of  the  method  of  right  and  wrong  cases.  Cattell 
stands  in  his  demonstration  entirely  on  the  ground  of  the  theory 
of  errors  of  observation  and  he  does  not  go  beyond  it.  His 
arguments  prevail  against  the  criticisms  of  his  view,  some  of 
which  miss  entirely  the  point  of  his  argument,  e.  g.  the  experi- 
mental demonstration  that  one  may  make  the  difference  be- 
tween two  light  intensities  so  small  that  it  cannot  be  perceived. 
It  seems  that  the  source  of  these  difficulties  is  the  introduction 
of  the  distinction  between  correct  and  wrong  judgments.  This 
is  a  logical  category.  What  is  immediately  given  are  not  right 
and  wrong  judgments  but  the  judgments  "greater",  "equal" 
and  'smaller";  the  correctness  or  incorrectness  of  the  judgments 
is  a  secondary  feature.  By  introducing  this  distinction  between 
correct  and  incorrect  judgments  it  becomes  necessary  to  dispose 
in  some  way  of  the  equality  cases,  which  as  a  matter  of  fact  were 
so  troublesome  a  feature  in  the  method  of  right  and  wrong  cases, 
that  some  investigators  have  tried  to  get  rid  of  them  by  not 
allowing  the  subject  to  pass  the  judgment  "equal."     The  imme- 

*G.  S.  FuLLERTOX  and  J.  McKeen  C.\TTELL,  On  the  Perception  of  Small 
Differences,  1892,  p.  12  sqs. 


46  PROBLEMS  OF  PSYCHOPHYSICS 

diate  data  of  experiments  for  the  determination  of  the  sensitivity 
of  a  subject  are  the  observed  frequencies  of  the  cases  in  which 
the  judgments  "greater",  "equal"  and  "smaller"  were  passed. 
We  will  try  to  base  our  judgment  on  the  sensitivity  of  a  subject 
on  these  percentages  for  various  differences,  without  obliterating 
one  feature  of  the  results  by  eliminating  a  class  of  judgments, 
and  without  introducing  the  logical  category  of  right  and  wrong 
judgments.  We  shall  use  for  this  purpose  the  notion  of  the 
probability  of  a  judgment  of  certain  type,  which  was  introduced 
in  the  preceding  chapters. 

Let  us  suppose  a  subject  compares  n  pairs  of  stimuli  which 
have  one  stimulus,  the  standard,  in  common.  Let  the  order 
in  which  the  comparison  stimuli  are  presented  be  the  same  for 
all  the  pairs,  e.  g.  let  the  standard  be  the  first  stimulus  of  every 
pair  Let  us  call  the  different  stimuli  with  which  the  standard 
is  compared  r^,  r,,  ....  r^,  and  let  us  suppose  that  the  "pairs  are 
presented  in  such  an  order  that 

ri<r,< <r^. 

Keeping  all  the  conditions  which  might  possibly  influence  the 
judgment  as  constant  as  possible  we  give  this  series  repeatedly 
to  a  subject.  The  judgments  will  vary  between  "greater," 
"lighter"  and  "equal"  and  there  exists  under  the  conditions 
of  the  experiments  a  definite  probability  for  every  comparison 
resulting  in  a  judgment  of  one  of  the  three  types.  The  proba- 
bilities that  a  "greater "-judgment  will  be  given  may  be  called 

Pi,  Vi, Vn 

where  p^  is  the  probability  that  the  judgment  "greater"  will 
be  given  on  the  comparison  of  the  standard  with  the  comparison 
stimulus  of  the  k*^^  pair.  The  probabilities  that  a  "  greater  "- 
judgment  will  not  be  given  are  correspondingly 

1-/'   ='72 

l-?'n  =  '/n- 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  47 

The  cases  where  a  ''greater "-judgment  is  not  given  comprise 
those  cases  where  the  comparison  stimulus  was  judged  equal  to 
the  standard,  and  those  cases  where  it  seemed  to  be  smaller  than 
the  standard.  The  comparison  stimuli  may  be  chosen  in  such 
a  way  that  the  probability  of  a  "greater "-judgment  is  s  nail  for 
the  stimuli  at  the  beginning  of  the  series,  and  large  (close  to  the 
unit)  for  the  stimuli  at  the  end  of  it.  Presenting  this  series  to 
the  subject  we  w^ill  obtain  a  determination  of  the  just  perceptible 
positive  difference  in  the  comparison  stimulus  of  the  first  pair 
on  which  the  judgment  "greater"  is  given,  all  the  preceding 
pairs  being  judged  "smaller"  or  "equal."  The  probability 
that  the  k^*^  pair  is  the  first  to  be  judged  "greater"  i^  the  com- 
pound probability  that  the  comparison  stimulus  of  this  pair  is 
judged  "greater"  and  that  the  judgment  "greater"  is  not  passed 
on  any  one  of  the  pairs  w^th  smaller  comparison  stimuli.  The 
probability,  therefore,  that  r^.  will  ba  obtained  a^  a  dete  mination 
of  the  just  perceptible  difference  is  given  by 

^k  =  ^i92 9k-i  /'k  =  /'k    ^     <7i-     (!■) 

i  =  1 

This  quantity  is  different  for  every  pair.  In  a  series  in  which 
the  probabilities  of  "greater "-judgments  increase  from  very 
small  values  at  the  beginning  of  the  series  to  large  values,  the  P's 
increase  at  first  and  decrease  after  having  attained  a  maximum, 
if  certain  conditions  are  fulfilled  which  will  be  enumerated  later. 
The  results  of  N  determinations  of  the  just  perceptible  positive 
difference  after  being  brought  in  proper  order  will  have  this 
form: 

The  stimulus  Tj  was  the  first  to  be  judged  "greater"   N,    times, 

(I  li         „       ((  n         u  It  It  It  vr  (( 

1 2  i>l2 

The  stimulus  rj^  was  the  first  to  be  judged  "greater"  N^  times, 
where 

The  algorithm  of  the  method  of  just  perceptible  differences  pre- 
scribes to  take  the  arithmetical  mean  of  all  these  determinations 


48  PROBLEMS  OF  PSYCHOPHYSICS 

for  the  final  determination  of  the  just  perceptible  positive  differ- 
ence.    This  average  is  given  by 

In  a  considerable  number  of  determinations  every  stimulus  r^ 
will  tend  to  occur  in  a  number  of  times  proportional  to  its  prob- 
ability P]^.  The  most  probable  result  for  a  great  number  of 
determinations  of  the  just  perceptible  positive  difference  is  there- 
fore 

T=r,P,+  r,P,  +  ....+r^P^ (H.) 

This  relation  shows  that  the  result  of  the  method  of  just  percep- 
tible differences  depends  on  the  probabilities  of  the  judgments 
of  different  types  and  that,  therefore,  its  basis  is  identical  with 
that  of  the  error  methods.  The  second  remark  which  we  have 
to  make  regards  equation  I.     The  value  of  P  does  not  change 

if  the  order  of  the  terms  q^,  qj, qk.i    p^  is  changed,  because 

a  product  is  independent  of  the  order  of  its  factors.  It  does  not 
matter  either  if  a  stimulus  r^^^  is  given  before  r^,  because 
P]j  +  s  does  not  enter  into  relation  I.  This  means  that  the  prob- 
ability of  a  stimulus  being  the  smallest  on  which  a  "  greater  "- 
judgment  is  given  does  not  depend  on  the  order  in  which  the 
pairs  are  presented.  The  method  of  just  perceptible  differences, 
therefore,  is  not  tied  down  to  the  rule  of  presenting  the  stimuli 
in  the  order  of  their  magnitude. 

Two  interpretations  may  be  given  to  relation  II.  The  first 
is  based  on  the  notion  of  the  mathematical  expectation;  its  psychol- 
ogical bearing  is  less  obvious  but  it  is  moie  serviceable  for  cer- 
tain practical  purposes.  Multiplying  each  stimulus  with  the 
probability  that  it  will  be  obtained  as  a  result  of  the  method 
of  just  perceptible  differences  gives  what  is  called  the  mathematical 
expectation  for  this  stimulus  being  the  threshold,  and  the  sum 
of  these  products  for  all  the  pairs  of  the  series  gives  the  mathe- 
matical expectation  for  the  entire  series.  The  mathematical 
expectation  for  N  repetitions  of  the  same  event  is  N  times  that 
for  a  single  event  and  taking  the  results  of  many  observations 
for  its  determination  gives  the  result  a  greater  exactitude.  This 
interpretation  of  the  algorithm  of  the  method  of  just  percep- 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  49 

tible  differences  is  independent  from  any  particular  hypothesis 
as  to  the  law  of  distribution.  One  may  try  to  make  its  meaning 
clear  in  this  way.  Let  us  suppose  that  A  pays  to  B  a  sum  of 
money  proportional  to  the  intensity  of  the  comparison  stimulus 
which  is  obtained  as  a  determination  of  the  just  perceptible 
positive  difference,  how  much  must  B  pay  to  A  in  order  to  in- 
duce him  to  make  this  agreement  and  to  make  it  a  fair  wager? 
Relation  II  gives  the  answer  that  B's  payment  must  be  propor- 
tional to  T,  because  in  this  case  the  expectation  of  A  is  equal 
to  that  of  B. 

The  second  interpretation  is  based  on  the  signification  of  the 
arithmetical  mean  for  symmetrical  distributions.  Taking  the 
average  of  a  series  of  observations  means  that  one  tries  to 
determine  the  most  probable  value,  if  the  distribution  is  sym- 
metrical.    The  most  probable  result  of  one  determination  by  the 

series  t^,  rj, r^^  is  the  one  for  which  the  product   of  formula 

I  is  a  maximum.     Pj^  is  a  maximum  if 

or  introducing  the  corresponding  expressions 

i  =  A-  2  i  =  k-i  i  =  k 

P^..^      n       q,<P^      n       q,>P^  +  ,    n       q,....{lU.) 
i=l  i=i  i=i 

By  splitting  up  this  relation  into  two  and  eliminating  the  com- 
mon factors  we  obtain  the  conditions 

Pk.i<1k.iPk     and     Pi,>qi,Pi,^^ 
which  are  identical  with 

-^^^-<Kand    ij^>h..      (IV.) 

It  is,  furthermore,  a  fact  that  the  probability  of  a  "heavier "- 
judgment  becomes  the  greater  the  smaller  negativ3  differences, 
and  the  greater  positive  differences  of  the  stimuli  become,  so  that 
relation  IV  must  be  simultaneous  with 

Pk-i  <  Pk<  Pk  +  i (IVa.) 

This  relation  shows  that  the  position  of  the  maximum  of  P;^ 
depends  on  the  probability  of  a  "h3avier "-judgment  for  this 


50  PROBLEMS  OF  PSYCHOPHYSICS 

pair  and  on  those  for  the  pairs  immediately  preceding  and  im- 
mediately following.  The  formulae  IV  and  IVa  give  the  condi- 
tions, that  the  value  of  a  certain  Pj^  is  greater  than  those  in  its 
neighborhood.  If  these  conditions  are  fulfilled  it  does  not  fol- 
low that  Pk  is  the  absolute  maximum  (i.  e.  greater  than  any 
other  value  of  the  series),  but  if  Pj^  is  the  absolute  maximum 
the  conditions  IV  and  IVa  must  be  fulfilled.  Let  us  suppose 
that  the  first  k  stimuH  i\,  i\, r^  are  chosen  in  such  a  way  that 

P,<P,<...<P^ 

and  let  us  consider  the  possible  effect  of  our  choice  of  the  next 
stimuli.     Does  it  depend  on  our  choice  of  the  stimuli  rj,  ^  i,  i\  +2 

r^  that  Pj^  is  the  maximum,  or  is  the  peculiarity  of  i\  having 

the  greatest  probability  of  being  observed  as  the  just  perceptible 
difference  based  on  some  particular  quality  of  this  stimulus? 
In  the  latter  case  we  will  have  in  our  result  a  determination  of 
the  just  perceptible  difference  irrespectively  of  the  way  in  which 
we  approach  it,  but  in  the  first  case  this  quantity  would  be  differ- 
ent for  different  series  of  comparison  stimuli.  We  suppose  that 
the  probabilities  of  "greater "-judgments  are  analytic  func- 
tions of  the  intensities  of  the  comparison  stimuli,  so  that  we  have 


Pn='y(^-n) 


and  we  choose  the  stimulus  rj^+j  only  slightly  different    from 
v^,  so  that  the  higher  powers  of  the  difference 

''k-fi-''k=o 
can  be  neglected.     The  second  part  of  relation  IV  has  the  form 


l-^(^w) 


The  value  of  W  (x)  is,  by  its  nature  of  being  a   mathematical 
probability,  smaller  than  1,  and  the  term  on  the  left  side  of  the 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  51 

above  relation  is  the  sum  of   a  geometric  series.      Developing 
4'(rj£+J)  in  a  power  series  we  find 

1    •  ^   ■ 


or 


0' 


Neglecting  the  powers  of  3  and  summing  up  the  series  we  find 

d<: ^ (\.) 

[i-^'(^k)]^'('-k) 

as  the  condition  with  which  we  must  comply  in  our  choice 
of  the  stimulus  t^.^^  in  order  to  make  Pk  +  i  smaller  than  Pj^. 
The  terms  ^(rj^)  and  l-^'frj^)  are  necessarily  positive  num- 
bers and  ^t''(rij)  is  also  positive,  because  ^"^(r)  is  an  increasing 
monoton  function;  d  is,  therefore,  positive.  From  this  it  fol- 
lows that  it  is  always  possible  to  choose  a  stimulus  rj^  +  j  greater 
than  rj.  in  such  a  way  that  r^^  is  more  likely  to  come  out  as  the 
result  of  the  determination  of  the  just  perceptible  difference  than 
r^  4. , .  This  may  be  done  by  choosing  r^  + 1  only  slightly  diff- 
erent from  rj.. 

It  is  obvious  that  we  may  confine  our  considerations  of  the 
conditions  for  P^  +  1  >  Pk  to  the  caseT(r;^)<i,  because  P)t-^i 
must  necessarily  be  smaller  than  Pj^  if  ^"(r^)  is  greater  than  ^. 
We  put 

and  find  that  P^  ^  ^  will  be  greater  than  Pj^  if 

or 

a+e)T(r,^J>(^-e) 

which  determines  the  relation 

^■(^k  +  x)>  i^ (V-) 

^'^^^        1  +  2^ 

This  fraction  is  smaller  than  1  and  it  may  represent,  therefore, 
a  mathematical  probability.     It  remains  to  show  that  it  may 


52  PROBLEMS  OF  PSYCHOPHYSICS 

represent  the  probability  of  a  "greater "-judgment  on  one  of 
the  stimuli  which  may  be  chosen  as  the  k+1  pair  of  our  series. 
These  stimuli  have  to  satisfy  the  relation 

We  find  for  the  greatest  value  of  the  difference  ^(rj^)-^(rjj^J 
l-2e  {l-2ey 


\-e- 


l+2e  2(l  +  2g) 


which  is  always  negative.  From  this  it  follows  that  it  is  always 
possible  to  find  a  stimulus  rjj  +  j>rj^  which  satisfies  the  relation 
V.  as  long  as  p<i.  The  stimulus  determined  in  this  way  is 
more  likely  to  be  observed  as  a  result  of  a  determination  of  the 
threshold  by  the  method  of  just  perceptible  differences  than  the 
preceding  stimulus.  • 

These  considerations  show  the  importance  of  the  choice  of  the 
comparison  stimuli.  The  generality  of  our  discussion  is  not 
impaired  by  the  supposition  that  there  is  only  one  maximum 
of  the  P's,  because  if  there  are  several  values  which  satisfy  rela- 
tion IV  we  will  have  to  consider  an  intermediate  value  (the  mean) 
and  the  influence  of  our  choice  of  the  comparison  stimuli  is  equal 
to  the  sum  of  the  disturbances  in  the  position  of  the  single  maxima. 
The  practical  bearing  of  this  part  of  the  demonstration  is  that 
one  must  not  pick  out  the  comparison  stimuli  at  random,  but 
one  will  choose  them  according  to  a  principle.  The  most  obvious 
rule  will  be  to  choose  equidistant  values 

ri  =  r, 
r,  =  r^  +  d 
r3  =  ri  +  2d 


r„  =  ri+(n-l)d. 


The  choice  of  d  will  depend  on  the  accuracy  aimed  at  in  the  de- 
termination  of   the   threshold. 

The  outcome  of  a  definite  series  of  experiments  for  the  deter- 
mination of  the  just  perceptible  difference  is  a  well  defined  quant- 
ity, but  whether  the  final  average  has  the  signification  of  a  most 
probable  result  depends  on  the  distribution  of  the  observed  values. 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  53 

The  discussion  of  this  question  will  show  that  this  suppos  tion 
may  be  made  under  certain  conditions.  The  result  of  the  method 
of  just  perceptible  differences,  therefore,  may  have  a  significa- 
tion independent  from  the  particular  series  of  experiments  by 
which  it  is  reached. 

The  arithmetical  mean  is  the  most  probable  value  of  a  set  of 
observations,  if  the  distribution  is  symmetrical  i.e. if  positive  and 
negative  deviations  have  the  same  probability.  The  <t>(^)-law 
is  only  a  special  case  of  symmetrical  distributions.  Under  what 
conditions  will  the  values  i\  Nj^.  be  distributed  symmetrically 
around  their  mean?  The  nature  of  this  distribution  obviously 
depends  on  the  values  of  the  P's.  The  P's  are  constituted  of 
the  p's  and  of  the  q's  of  the  pairs  and  since  it  depends  on  our 
choice  which  pairs  we  will  use,  no  a  priori  statement  is  possible 
in  regard  to  any  particular  series  unless  one  knows  the  pairs  and 
the  respective  probabilities  of  "h"  judgments.  Generally  the 
distribution  will  not  be  symmetrical,  but  it  may  very  well  be 
that  a  particular  series  has  a  symmetrical  distribution  or 
one  which  approaches  this  type.  A  priori  one  even  cannot  make 
the  supposition  that  the  distribution  is  regular  i.  e.  that  it  shows 
an  uninterrupted  increase  at  first  and,  after  having  attained  a 
maximum,  an  uninterrupted  decrease.  Indeed,  if  the  compar 
ison  stimuli  are  picked  out  entirely  at  random  it  may  very  well 
be,  that  the  P's  increase  in  value  at  first,  then  after  having  reached 
a  secondary  maximum  decrease  and  increase  again  later  on.  This 
will  be  the  case  if  there  are  in  the  series  two  or  more  comparison 
stimuli  with  only  slightly  different  probabilities,  which  are 
smaller  than  ^,  for  the  appearance  of  a  "'greater- "judg- 
ment. By  taking  the  stimuli  in  equal  intervals  one  is  to 
some  extent  guarded  against  this  eventuality,  but  if  one  was 
unfortunate  in  the  choice  of  the  comparison  stimuli  one  cannot 
eliminate  this  influence  by  any  amount  of  care  in  the  perform- 
ance of  the  experiments,  or  by  the  combination  of  any  number 
of  observations.  It  is  not  possible  either  to  eliminate  the  influ- 
ence of  a  skew  distribution  by  taking  the  arithmetical  mean  of 
a  great  number  of  experiments.  On  account  of  the  fact  that  one 
cannot  make  any  statement  about  the  distribution,  it  is  impos- 
sible to  say  whether  the  final  result  of  an  individual  series  of 


54  PROBLEMS  OF  PSYCHOPHYSICS 

experiments  by  the  method  of  just  perceptible  differences  has 
the  character  of  being  the  most  probable  value  of  the  threshold 
or  not. 

The  aspect  of  the  problem  is  entirely  different  if  one  has  to 
deal  not  with  the  results  of  one  series  of  comparison  stimuli, 
but  with  the  results  of  several  series  with  different  arrangements 
of  the  pairs  of  comparison  stimuli.  Owing  to  the  beautiful 
theorem  which  Bruns  calls  the  conservation  of  the  ^  {■jr)-type 
this  mixture  of  independent  distributions  tends  to  produce  the 
(p  ( ^)  -distribution.  We  have  therefore  to  deal  in  this  case 
with  a  symmetrical  distribution  and  the  arithmetical  mean  is 
the  most  probable  value.  In  the  apparently  trivial  caution  "not 
to  approach  the  threshold  always  by  the  same  steps"  lies  to  a 
large  extent  the  justification  of  the  algorithm  of  the  method  of 
just  perceptible  differences.  Most  investigators,  as  a  matter  of 
fact,  used  different  series  of  comparison  stimuli  for  their  deter- 
minations of  the  threshold,  and  this  mixture  of  distributions 
produced  a  symmetrical  distribution  which  gave  the  results  a 
signification  independent  of  the  comparison  stimuli  which  had 
been  used.  One  might  have  surmised  that  the  results  of  the 
method  of  just  perceptible  differences  must  have  a  definite  signifi- 
cation l^ecause  the  results  of  different  investigators  are  in  gen- 
eral agreement  in  spite  of  large  individual  differences.  It  is 
hardly  imaginable  that  a  regularity  like  Weber's  Law  could 
have  been  found  and  verified  by  independent  observations,  if 
every  single  investigator  had  determined  another  quantity,  as 
would  have  been  the  case  if  the  results  depended  essentially  on 
the  choice  of  the  comparison  stimuli.  A  very  important  prac- 
tical caution  for  the  application  of  the  method  of  just  perceptible 
differences  follows  from  these  considerations:  Do  not  use  always 
the  same  series  of  comparison  stimuli,  but  use  as  many  different 
arrangements  as  can  be  done  conveniently.  In  an  extended 
series  of  observations  one  will  go  about  systematically  in  choosing 
the  arrangements.  In  agreement  with  the  procedure  suggested 
above  one  will  use  a  series  of  stimuli  of  the  same  difference, 
but  instead  of  beginning  with  i\  one  will  use  a  stimulus  slightly 
different,  say  r',  so  that  the  second  arrangement  will  be 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  55 

r'3  =  r\  +  2d 


r',  =  r\+(n-l)d 

If  one  intends  to  use  k  different  arrangements  one  will  choose 
the  stimuli  so  that  no  point  of  the  interval  is  favored  which 
can  be  done  by  making  r',-r^  =  ^. 

If  we  have  succeeded  in  choosing  d  in  such  a  way  that  none  of 
the  pairs  complies  with  relation  V  (which  can  be  done  by  choosing 
d  not  too  small)  the  P's  will  increase  at  first  up  to  a  certain  maxi- 
mum, which  is  reached  for  the  stimulus  rj^,  and  the  value  of  r  for 
which  p  =  ^  must  be  in  one  of  the  intervals  r^_^  i\  or  r;.  i\  +  ^.  If 
we  have  chosen  our  comparison  stimuli  in  such  a  way  as  not  to 
favor  any  value,  there  will  be  an  even  probability  that  the  stimulus 
for  which  P  is  a  maximum  is  greater  or  smaller  than  the  stimulus 
for  which  p  =  i.  From  this  it  follows  that  in  a  large  number 
of  determinations  of  the  just  perceptible  positive  difference  made 
with  different  series  of  comparison  stimuli,  the  most  probable 
result  is  given  by  that  amount  of  difference  for  which  there  exists 
the  probability  h  that  the  judgment  "greater"  will  be  given. 

Relation  IV  explains  the  surprising  fact  that  the  just  percep- 
tible difference  seems  to  become  smaller,  if  new  comparison  stim- 
uli are  interpolated.  The  result  of  such  an  interpolation  may 
be  as  follows:  As  a  rule  this  interpolation  will  be  made  system- 
atically and  one  will  not  only  not  avoid  interpolating  new  pairs 
in  the  neighborhood  of  the  point  where  one  expects  to  find  the 
threshold,  but  one  will  be  careful  to  do  so  since  this  is  the  region 
in  which  one  is  most  interested.  This  interpolation  of  new  pairs 
may  produce  a  shifting  of  the  most  probal^'.e  value,  which  will 
no:  be  the  same  in  the  new  series  as  it  was  in  the  first.  In  regard 
to  this  change  the  following  rule  holds:  If  the  most  prob- 
able result  varies  it  must  become  smaller  necessarily.  Indeed, 
let  there  be  three  stimuli  rj..,,  r;^  and  rj^.^,  for  wliich  the  prob- 
abilities of  "greater "-judgments  are  Pi^.,,  p;^  and  Pk  +  i-  The 
probabilities  that  one  of  these  stimuli  will  be  obtained  as  a  deter- 


56  PROBLEMS  OF  PSYCHOPHYSICS 

miiiation  of  the  just  perceptible  difference  are  P^.,,  P^  and  P^^., 
respectively.     If  there  exists  the  relation 

■Pk-i<^k>-Pk  +  i 

we  also  must  have 


^k-i 


<p^  and    -- — —  >Pk  +  i- 


Now  let  us  interpolate  a  hew  stimulus  r'j^  between  r^.,  and  t^ 
and  another  stimulus  r'i^_^i  between  rj^  and  r;^.^!  so  that 

^k-i  <  ''k<  'k<»'k  +  i<  ^k  +  i 

where  p'^  and  p'k  +  i  stand  for  the  probabilities  of^  "greater"- 
judgments  on  the  new  stimuli.  It  follows  from  p'k  +  i<Pk  +  i 
that  there  must  be  also 

>Pk  +  i- 


l-Z'k 


This  means  that  the  position  of  the  maximum  is  not  changed 
by  interpolating  new  stimuli  between  the  stimulus  for  which  P  is 
a  maximum  and  the  subsequent  stimulus.  This  fact  is  not  sur- 
prising since  no  stimulus  higher  than  rj^  enters  into  the  formula 
for  Pj..      No  general  statement  is  possible  in  regard  to  the  relations 

_%—  >/>!,  and  p\<p^., 
^-Pk-i 

and    one    sees    that    there    may    be    ~ —  >  p'     as    well   as 

l-Z'k-i 

Fk-i    <^^^^.     It  must   be    decided    by  formula  V  which    of  the 
1-^k-i 

values  Pi-,  Pk'  and  P^  is  the  maximum.  It  follows  from 
these  considerations  that  the  most  probable  result  for  the  determi- 
nation of  the  just  perceptible  difference  may  be  changed  by  the 
interpolation  of  new  stimuli,  and  if  it  is  changed  it  must  become 
smaller.  This  indicates  that  there  is  a  certain  danger  in  taking 
intermediate  steps  which  are  very  small  and,  generally  a  little 
vaguely   speaking,   one    ought    not    to   take    intervals  which  are 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  57 

too  small.  It  will  be  see  later  that  the  size  of  the  interval  de- 
pends on  the  number  of  experiments  one  is  willing  to  make  and 
that  there  exists  a  principle  by  which  one's  choice  of  the  intervals 
may  be  guided. 

Now  let  us  suppose  that  the  interpolation  of  new  stimuli  was 
carried  through  in  such  a  way  as  not  to  change  the  position  of 
the  maximum  of  the  P's,  but  that  the  number  of  stimuli  preced- 
ing this  maximum  has  been  increased.  This  will  have  the  effect 
of  adding  some  more  factors  to  the  product  in  formula  I,  and 
since  every  factor  is  smaller  than  the  unit  the  values  of  the  P's  in 
the  new  series  must  be  smaller  than  those  in  the  first,  series.  This 
means  that  it  is  less  likely  that  an  individual  result  will  give  the 
most  probable  result,  which  stands  out  less  distinctly  in  the 
new  series  than  it  did  in  the  first  one  and  the  probability  of  smaller 
results  has  increased   correspondingly. 

■  These  considerations  show  that  the  effect  of  the  interpolation 
of  new  pairs  of  comparison  stimuli  must  necessarily  be  a  more 
or  less  considerable  diminution  of  the  just  perceptible  difference. 
By  this  it  is  not  intended  to  say  that  all  the  observed  diminutions 
of  the  just  perceptible  difference  are  entirely  due  to  this  circum- 
stance, because  there  is  some  evidence  that  the  attitude  of  the 
subject  is  not  quite  the  same  in  judging  differences  in  a  series 
with  small  intervals  than  it  is  in  series  with  large  intervals.  The 
study  of  the  influence  of  such  a  variation  of  the  conditions  on 
attention  is  a  separate  problem,  but  our  considerations  show 
that  the  interpolation  of  new  pairs  of  comparison  stimuli  may 
have  the  effect  of  diminishing  the  just  perceptible  difference, 
even  if  the  subject  is  not  influenced  at  all  by  the  new  condi- 
tions. The  lowering  of  the  threshold  is  not  only  not  an  argu- 
ment for  the  ruling  out  of  a  particular  series,  but  one  would  have 
to  be  suspicious  of  a  change  in  the  attitude  of  the  subject,  if  the 
interpolation  of  new  pairs  would  constantly  fail  to  have  the 
effect  described. 

Until  now  w'e  have  confined  our  discussion  to  the  just  percep- 
tible positive  difference.  The  complete  method  of  just  percep- 
tible differences  requires  the  determination  of  three  other  quanti- 
ties: The  just  imperceptible  positive  difference,  the  just  percep- 
tible  negative   difference    and   the   just   imperceptible   negative 


58  PROBLEMS  OF  PSYCHOPHYSICS 

difference.  All  these  quantities  are  expressed  by  formulae 
analogous  to  formula  II,  so  that  the  theoretical  discussions  based 
on  this  relation  are  valid  for  all  the  quantities  in  question. 

Presenting  our  series  of  pairs  of  stimuli  in  the  order  r^^,  r^^.i, 

r,,  where  T^>r^_i> >r2>ri  we  will  have  a  determination 

of  the  just  imperceptible  positive  difference  in  the  first  stimulus 
on  which  the  judgment  "greater"  is  not  given,  all  the  preced- 
ing pairs  being  judged  "greater".  Using  the  same  denotation 
as  before  we  find  for  the  probability  that  the  stimulus  r^  will  be 
obtained  as  a  determination  of  the  just  imperceptible  positive 
difference 

because  p^  Pn.i  Pk  +  i  is  the  probability  that  on  all  the  stim- 
uli greater  than  i\  a  "greater "-judgment  will  be  given  and  q^ 
is  the  probability  that  such  a  judgment  will  not  be  given  on  r^. 
These  probabilities  F^'  are  different  for  the  different  stimuli 
and  in  a  great  number  of  experiments  the  most  probable  result 
is  that  every  stimulus  will  be  obtained  as  a  determination  of  the 
just  imperceptible  positive  difference  in  a  number  of  times  pro- 
portional to  this  probability.  The  most  probable  result  of  these 
determinations  of  the  just  imperceptible  positive  difference  is 
therefore  ' 

r  =  }\P\  +  r,P',  + +  >\,P'^       (VII.) 

Formulae  VI  and  VII  are  analogous  to  I  and  II.  If  the  stimu- 
lus rj,  has  the  greatest  probability  for  being  obtained  as  a  result 
of  a  determination  of  the  just  imperceptible  positive  difference 
we  must  have 

or  introducing  the  expressions  for  these  probabilities 

PnPn-l Pk%.l<PnPn-l Pk  +  l(lk>  Pn  Pn-1 Pk  +  2  (Jk  +  1 

which  are  identical  with 

Pk%-i<9k   and  /^k  +  i9k>%  +  i      (VIII.) 

The  relations  VIII  and  formula  VI  may  be  used  for  showing 
that  the  result  of  a  particular  series  depends  essentially  on  our 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  59 

choice  of  the  comparison  stimuli,  but  that  it  is  independent  from 
the  order  in  which  the  pairs  are  presented.  This  demonstra- 
tion requires  only  a  repetition  of  our  above  considerations  in- 
troducing the  p's  in  the  place  of  the  q's  and  vice  versa.  We  will 
consider  only  the  effect  of  an  interpolation  of  new  pairs  of  com- 
parison stimuli  in  that  part  of  a  series  where  the  maximum  of 
F'y.  is  situated.  Let  us  suppose  we  have  three  stimuli  Tj^.i, 
Tjj  and  ri^^i  for  which  the  probabilities  of  "greater "-judgments 
are  Pi^.i,  pj^  arill  p^  +  i,  and  let  their  probabilities  of  being  ob- 
served as  results  of  the  just  imperceptible  positive  difference  be 
P'k-i>  P'k  a^d  P'k  +  i  respectively.  If  F'^  is  a  maximum  we 
must  have  the  relations 

g,,<  -^^    and  q^>   J^±i-. 
1-^k  l-9k  +  i 

as  it  follows  from  the  relations  VIII.  Now  let  us  interpolate  a 
new  stimulus  r'l^.i  between  t-^_j^  and  r,^  and  another  stimulus 
r'k  +  i   between  r^^   and  rj^  +  j,  so  that  we  have 

^k-i<^''k.i<''k<^'k  +  i<'k  +  i 
and,  therefore,  also 

/'k-i<n-i</'k<n  +  i</'k  +  i      (IX.) 

where  p'j^.^  and  p'^  +  i  designate  the  probabilities  of  a  "greater"- 
judgment  on  the  comparison  of  the  stimuli  t\_i  and  r'j^^i 
with  the  standard.  The  probabilities  of  "greater "-judgments 
and  the  probabilities  of  those  judgments  which  are  not  "greater"- 
judgments  are  in  the  relation  q  =  1-p  and  it  follows  from  IX  that 

9k-i  >  ^'k-i  >  '/k  >  9'k  +  1  >  'Zk  +  1 
We  may  conclude  from  this  relation  that 

o'     <    ^'^ 
^I  k-i^- 

l-9k 
but  we  cannot  conclude  that 

9k  >    '^'^^^ 


1-9'k  +  i 

From  this  it  follows  that  the  interpolation  of  new  pairs  of  com- 
parison stimuli  has  no  effect  on  the  most  probable  value  of  the 


60  PROBLEMS  OF  PSYCHOPHYSICS 

determination  of  the  just  imperceptible  positive  difference,  if 
stimuli  are  interpolated  which  are  smaller  than  the  stimulus 
which  has  the  greatest  probability.  The  interpolation  of  inten- 
sities which  are  larger  than  the  most  probable  result  in  the  first 
series  may  or  may  not  have  an  effect  on  the  situation  of  the 
maximum  of  probabilities;  if  the  position  of  the  maximum  of 
probabilities  has  been  changed  in  the  new  series  it  must  have 
shifted  towards  higher  values.  These  considerations  bear  out 
the  fact,  which  was  observed  before,  that  the  result  of  a  determin- 
ation of  the  threshold  by  the  method  of  just  perceptible  differ- 
ences depends  somewhat  on  the  size  of  the  intervals  which  are 
used.  It  is,  therefore,  not  necessarily  a  sign  of  incomplete  train- 
ing of  the  subject  or  of  his  inability  to  direct  his  attention  on  the 
comparison  of  the  stimuli,  if  series  with  small  differences  fail 
to  give  the  same  result  as  series  with  large  differences.  We 
supposed  in  our  demonstration  that  the  new  series  with  smaller 
intervals  contained  all  the  stimuli  of  the  old  series  and  some  new 
ones  in  addition,  whereas  one  frequently  will  use  series  of  differ- 
ent stimuli.  This  case  was  not  considered,  because  it  is  not 
possible  to  make  a  general  statement  about  the  comparative 
results  of  two  series,  unless  some  other  data  are  at  hand.  Our 
method  has,  furthermore,  the  advantage  of  showing  the  imme- 
diate effect  of  the  interpolation  of  new  stimuli. 

It  depends  on  the  distribution  of  the  V\  whether  the  algorithm 
of  the  method  of  just  perceptible  differences  leads  to  the  deter- 
mination of  the  most  probable  value  for  the  just  imperceptible 
positive  difference  or  not.  Also  in  this  case  it  is  indispensable  to 
have  the  results  of  a  great  number  of  different  series  in  order  to 
warrant  the  hypothesis  of  a  symmetrical  distribution.  The  arith- 
metical mean  has  the  character  of  the  most  probable  value  if 
this  condition  is  complied  with.  The  formula  for  P'j^  is  exactly 
analogous  to  the  formula  for  F^,  the  p's  and  q's  being  inter- 
changed. By  repeating  our  considerations  one  can  show  that 
P'ij  is  a  maximum  independent  of  our  choice  of  the  following 
stimuli  if  qk  =  i  or  since  Pk  +  '7k  =  lj  i^  Pk  =  i-  '^^^  stimulus 
for  which  Pk  =  i  is  the  most  probable  result  for  the  determina- 
tion of  the  just  perceptible  difference,  and  we  see  that  the  most 
probable   results  for  the   determination  of  the  just  perceptible 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  61 

and  of  the  just  imperceptible  positive  difference  are  identical. 
Tiie  determination  of  the  just  perceptible  positive  difference 
is  based  on  the  probabilities  of  the  stimuli  smaller  than  r^  and 
the  determination  of  the  just  imperceptible  positive  difference 
is  based  on  the  stimuli  beyond  this  value.  One,  therefore,  niust 
expect  to  find  differences  between  the  results  for  these  values 
for  every  particular  series  of  experiments.  Taking  the  average 
of  the  just  perceptible  and  of  the  just  imperceptible  positive 
difference  for  the  final  determination  of  the  threshold  in  the 
direction  of  increase,  has  the  signification  of  basing  one's  deter- 
mination of  the  difference  for  which  there  exists  the  probability 
i  that  a  "greater "-judgment  will  be  given  on  all  the  results  of 
the  series,  and  not  only  on  the  results  for  the  stimuli  in  the 
lower  part  of  the  series,  as  the  determination  of  the  just  percep- 
tible difference  does,  or  on  those  of  the  upper  part,  as  the  deter- 
mination of  the  just  imperceptible  difference  does. 

For  the  determination  of  the  just  perceptible  and  the  just 
imperceptible  negative  difference  we  have  to  consider  the  prob- 
abilities of  "smaller "-judgments.     Designating  by 

Wi>«2> >u^ 

the  probabilities  that  in  the  comparison  of  the  standard   with 

the  stimuli  r^,  r^,    r,^  a  "smaller "-judgment    will  be  given 

we  have 

1 — u^  =  z\ 

1 — U.^  =  V2 


for  the  probabilities  that  a  "smaller  "-judgment  will  not  be  given. 
The  stimulus  r^  is  a  determination  of  the  just  perceptible  neg- 
ative difference,  if  all  the  stimuli  greater  than  rj^  were  judged 
"greater"  or  "equal"  and  i\  is  judged  "smaller".  The  prob- 
ability of  this  compound  event  is 

f^k  =  i''nVl ^"k  +  lWk (X-) 


62  PROBLEMS  OF  PSYCHOPHYSICS 

The  most  probable  result  of  a  great  number  of  determinations 
of  the  just  perceptible  negative  difference  is 

S  =  r,U,^r,lJ,  + +  rJJ^      (XL) 

The  just  imperceptible  negative  difference  is  defined  as  the  small- 
est intensity  on  which  a  judgment  is  given  which  is  not  a 
"smaller "-judgment,  all  the  stimuli  of  smaller  intensities  being 
judged  "smaller."  The  probability  that  the  stimulus  rj^  will 
be  obtained  as  a  determination  of  the  just  imperceptible  nega- 
tive difference  is  equal  to  the  compound  probability  that  all  the 
stimuli  of  smaller  intensity  are  judged  "smaller"  and  that  on 
this  stimulus  a  judgment  is  given  which  is  not  a  "smaller "-judg- 
ment.    This  gives 

U\  =  u^u^....U],_^i\ '. (XII.) 

The  most  probable  result  of  a  great  number  of  observations  on 
the  just  imperceptible  negative  difference  is 

S'  =  r,U\  +  r,U\  + +  ;,t/',      (XIII.) 

Formula  X  and  XII  are  analogous  to  those  which  we  obtained 
for  the  probabilities  of  a  stimulus  being  observed  as  a  result  of 
the  determination  of  the  just  perceptible  and  of  the  just  imper- 
ceptible positive  difference.  One  may  find  from  these  formulae 
the  relations  which  must  be  satisfied  if  a  stimulus  v^  has  a  max- 
imum of  probability.  The  investigation  of  these  relations  leads 
to  the  conclusion,  that  the  theoretical  values  of  the  just  percep- 
tible and  of  the  just  imperceptible  negative  difference  are  the 
same.  The  just  perceptible  negative  difference  is  that  intensity 
of  the  comparison  stimulus  for  which  there  exists  the  probabil- 
ity \  that  a  "smaller "-judgment  will  be  given.  The  just  imper- 
ceptible negative  difference  is  that  intensity  for  which  there 
exists  the  probability  ^  that  a  "smaller "-judgment  will  not  be 
given.  The  average  of  these  quantities  is  a  more  accurate  de- 
termination of  the  intensity  for  which  there  exists  the  probabil- 
ity \  for  a  "smaller "-judgment. 

The  result  of  the  method  of  just  perceptible  differences  has, 
therefore,  a  meaning  perfectly  well  definable  in  terms  of  the 
probabilities   of  judgments   of   certain   type.     We   admit   three 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  63 

types  of  judgments.  There  is  a  certain  realm  for  the  intensities 
of  the  comparison  stimuli  inside  of  which  neither  the  judgments 
"greater"  nor  the  judgments  "smaller"  have  a  probability 
greater  than  the  sum  of  the  two  other  types  of  judgments.  Out- 
side of  this  realm,  however,  one  of  these  judgments  has  a  prob- 
ability exceeding  ^. 

We  have  explained  the  reasons  which  suggest  the  adoption 
of  the  method  of  irregular  variation.  The  results  obtained  by 
this  method  can  be  worked  out  by  the  method  of  just  perceptible 
differences,  if  one  takes  care  to  record  all  the  judgments  given 
and  if  all  the  comparison  stimuli  are  presented  to  the  subject  in 
one  round  i.  e.  that  a  pair  is  not  presented  a  second  time  before 
all  the  other  pairs  were  presented.  From  the  records  one  picks 
out  the  stimulus  which  is  the  smallest  on  which  a  "greater  "- 
judgment  was  given  and  the  largest  on  which  such  a  judgment 
was  not  given.  The  first  serves  for  the  determination  of  the 
just  perceptible  positive  difference,  the  second  for  the  determin- 
ation of  the  just  imperceptible  positive  difference.  The  next 
step  is  to  find  the  greatest  stimulus  on  which  a  "smaller "-judg- 
ment is  given  and  the  smallest  stimulus  on  which  such  a  judg- 
ment is  not  given;  these  stimuli  are  determinations  of  the  just 
perceptible  and  of  the  just  imperceptible  negative  difference. 
The  complete  records,  therefore,  may  be  worked  out  by  the 
method  of  just  perceptible  differences  and,  since  they  give  the 
percentages  of  correct  and  wrong  judgments,  also  by  any  one 
of  the  error  methods.  The  necessity  of  recording  all  the  judg- 
ments given  is  avoided  in  that  form  of  the  method  of  just  percep- 
tible differences,  which  is  recommended  by  most  writers,  where 
the  outcome  of  an  entire  series  is  reduced  to  one  number.  This 
little  saving  of  clerical  work  is  quite  inconsiderable  when  com- 
pared with  the  work  actually  spent  in  experimenting,  and  it  is 
by  far  outweighed  by  the  possibility  of  impairing  the  results 
through  the  influence  of  expectation  on  the  part  of  the  subject. 
In  the  complete  records  one  also  has  data  which  may  serve  other 
purposes  than  the  determination  of  the  threshold.  The  possi- 
bility of  presenting  the  pairs  in  different  order  in  every  round, 
furthermore,  gives  a  means  of  eliminating  influences  due  to  the 
arrangement  of  the  series. 


64  PROBLEMS  OF  PSYCHOPHYSICS 

Under  the  supposition  that  the  judgment  on  one  pair  is  not 
influenced  by  the  judgments  on  any  of  the  preceding  pairs,  i.  e. 
that  the  order  in  which  the  pairs  are  presented  is  irrelevant,  the 
result  of  the  method  of  just  perceptible  differences  is  identical 
with  that  of  the  method  of  irregular  variation.  Both  methods 
determine  the  intensities  for  which  there  exists  the  probability 
^  that  on  the  comparison  with  the  standard  stimulus  the  judg- 
ments "greater"  or  "smaller"  will  be  given.  These  methods 
are  entirely  independent  of  any  hypothesis  as  to  the  law  of  distri- 
bution of  correct  and  wrong  cases,  and  this  is  a  great  superior- 
ity over  the  method  of  right  and  wrong  cases.  The  essential 
feature  of  the  error  methods  is  that  one  attempts  to  find  certain 
constants,  which  enable  one  to  find  the  relative  frequency  of 
correct  cases  for  any  difference  from  the  oljserved  relative  fre- 
quency of  these  judgments  on  one  difference  or  on  several  dif- 
ferences, as  it  was  suggested  by  Miiller  and  Titchener.  It 
makes  but  little  difference  whether  one  uses  for  this  purpose  the 
simple  law  of  Gauss,  or  Fechner's  double  sided  law  of  Gauss,  or 
whether  one  may  use  one  of  Pearson's  formulae  or  any  other  rela- 
tion, which  by  its  nature  is  fit  to  represent  a  law  of  distribution. 
Among  all  these  different  distributions  the  law  of  Gauss  must 
necessarily  retain  a  very  prominent  place  for  all  practical  pur- 
poses, not  only  because  it  depends  only  on  one  parameter,  but 
also  because  it  is  until  now  the  best  understood  of  all  the  laws 
of  distribution  and  finally,  because  it  does  not  suffer  from  little 
shortcomings  as  e.  g.  the  discontinuity  of  the  second  derivative 
in  Fechner's  generalisation  of  this  law.  The  method  of  right 
and  wrong  cases  may  be  characterized  as  a  method  of  interpo- 
lation under  the  supposition  of  a  particular  law  of  distribution. 
The  scope  of  this  method  is  by  far  more  ambitious  than  that  of 
the  method  of  just  perceptible  differences  or  of  the  method  of 
irregular  variation,  but  it  can  be  reached  only  .by  means  of  a 
definite  hypothesis  which  may  be  dispensed  with  for  either  one  of 
the  two  methods. 

It  still  remains  to  consider  what  the  limits  are  inside  of  which 
we  may  expect  the  result  of  the  method  of  just  perceptible  differ- 
ences with  a  given  probability.  It  seems  very  obvious  to  formu- 
late the  problem  in  this  way: 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  65 

The  result  r,  may  be  expected  with  the  probability  P^; 

a  ti       „  a       it  (I  ti  n  i(  T) 

'■2  ^2' 

The  result  r„  may  be  expected  with  the  probability  P^^. 

One  makes  N  trials  each  one  of  which  must  result  in  ri,  or  in  r,, 
or  ....  or  in  r^^.  Which  is  the  probability  that  the  sum  of  all 
the  results  will  be  S  ?  One  may  try  to  find  the  most  probable 
result  and  the  limits  within  which  one  may  expect  a  deviation 
with  a  given  probability,  just  in  the  same  way  as  one  derives 
these  quantities  for  Bernoulli's  theorem.  This  formulation  is 
nothing  but  a  generalisation  of  the  so-called  problem  of  Moivre. 
The  algebraical  difficulties  of  this  problem  soon  become  very  great 
and  it  does  not  seem  that  there  exists  at  present  a  general  solution. 
In  any  case  it  is  not  likely  that  the  solution  will  be  simple. 

The  second  way  of  formulating  the  problem  starts  from  the 
consideration  that  the  most  probable  outcome  of  a  series  of  N 
experiments  will  be  that  r^  occurred  NP^  times,  rj  occurred 
NPj  times  and  so  on,  and  that  the  theorem  of  Bernoulli  gives 
the  probabilities  for  the  deviations  from  these  most  probable 
results.  One  therefore  may  compute  the  probable  deviations 
and  find  the  probable  error  of  the  final  result  from  them. 
This  way  leads  to  a  result,  but  it  seems  to  be  the  most  elegant 
and  most  profitable  way  to  start  from  our  first  interpretation 
of  formula  II  and  to  apply  the  following  theorem  of  Tchebit- 
cheff.*  Let  x,  y,  z,  ....  be  any  magnitudes  which  may  assume 
different  values,  each  one  with  a  certain  probability,  and  let  the 
values  of  x  be 

Xj,  x,,   —  x^ 

to  which  correspond  the  probabilities 

Pv  Pi,  ••••  Pk 
so  that 

k 

I  P^  =  1; 

*TcHEBiTCHEFF,  Joumal  de  Liouville,  (2)  Vol.  12,  1867,  p.  177. 


66 


PROBLEMS  OF  PSYCHOPHYSICS 


in  the  same  way  let  the  values  of  y  be 

Uv  Vz,  ••••  ih 

with  the  probabilities 

Qi,  q2,  ••••  9i 
so  that 

and  so  on.     We  then  have  ^- 

k  1 

I  p^x.  =  a,    I  q^y^=b,    

1  1 

for  the  mathematical  expectations  or  averages  of   x,  y, 

k  1 


i'9k  =  l, 


and 


for  the  mean  values  of  the  squares.  Under  these  conditions 
the  theorem  holds,  that  the  probability  P  that  the  arithmetical 
mean 

^k  +  Vk+---- 


of  the  observed  values  of  x,  y,  z,  ...  is  contained  between  the  lim- 
its 

a  +  b+  ...  ._  1       /a,  +  6,+  .  .  ..  _a'  +  b'+  .... 
n  t     y  n  n 

and 


a  +  6+.... 


is  larger  than  1- 


+ 


IV^ 


1+&1+ 


a^  +  6?  + 


This    theorem  may   serve    for    the     determination   of  the   ac- 
curacy  of   the   results  of  the  method   of  just    perceptible  differ- 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  67 

ences.  Let  us  suppose  that  there  were  h  different  series  of 
comparison  stimuli,  the  first  arrangement  consisting  of  the 
comparison  stimuli 


which  have  the  probabilities  for  being  obtained  as  determina- 
tions of  the  just  perceptible  difference 

F^,  r^,  ....   "ki 

in  the  second  arrangement  the  stimuli 

r\,  r'„  ....  r\ 
were  used  which  had  the  probabilities 

pt       pr  pi 

r  .^f  r  2,  ■■■■  -r  ] 

respectiveh",   for  being  obtained   as   determinations   of  the  just 
perceptible  difference;  and  so  on.     We  then  have 

k  1 

1  I 

for  the  mathematical  expectations  in  the  single  series,  and 

k 

lP,r,^  =  a„   Ir\P\  =  h„.... 


for  the  mean  values  of  the  square.  With  the  first  arrangement 
we  make  n  experiments,  with  the  second  n'  experiments,  with 
the  third  n"  experiments,  ....,  so  that 

n  +  n'  +  n"+  ....=N. 

It  will  liave  the  same  effect  for  the  calculation,  if  we  take  the 
mean  value  and  the  mean  square  for  each  series  as  often  as  this 
series  was  given.  We  then  have  a  probability  P  that  the  arith- 
metical mean 

nr^  +  n'r'^.  +  .... 

N 


68  PROBLEMS  OF  PSYCHOPHYSICS 

of  the  observed  values  will  be  contained  between  the  limits 


fv- 


na  +  n'b+  ...         1    Jna^^n'b,+ .  .  .  .     na'  +  n'b'+. 


N  t    \  N  N 

and 


na  +  n'b+  ..  .._  1       La,  +  n'b^+ .  .  .  .  _na' +n'b'' +  .  .  .  . 
N  t    V  ^  ^ 

t^ 
which   is   greater  than    1  -  —  .     This    formula    shows    the   ad- 

N 

vantage  of  using  great  numbers  of  experiments,  because  the  dif- 

t^ 
ference  1- —    approaches  the  unit  if  N  increases  and  t  remains 

constant.  This  means  that  it  is  possible  to  choose  the  number 
of  experiments  so  large  that  one  may  expect  with- a  probability 
as  little  different  from  the  unit  as  one  pleases,  that  the  actual 
result  will  be  within  given  limits  from  the  calculated  result. 

In  regard  to  the  limits  of  accuracy  it  may  be  allowed  to  make 
the  following  comparison  of  the  method  of  just  perceptible  dif- 
ferences with  the  error  methods.  The  limits  of  accuracy  of  an 
empirical  determination  of  a  probability  depend  on  the  coeffi- 
cient of  precision  in  Bernoulli's  theorem,  which  is  a  function  of 
the  probability  of  the  event.  The  coefficient  of  precision  has 
a  minimum  for  the  probability  \,  so  that  events  which  may  be 
compared  with  the  tossing  up  of  a  coin  have  the  smallest  pre- 
cision. The  coefficient  of  precision  increases  as  the  value  of  the 
probability  of  the  event  approaches  zero  or  the  unit.  The  accu- 
racy of  the  method  of  just  perceptible  differences  depends  on 
empirical  determinations  of  the  P's,  which  are  constituted  of  the 
p's  which  are  used  in  the  error  methods.  Formula  II  shows 
that  the  P's  are  always  smaller  than  the  corresponding  p's  for 
the  same  stimulus  except  the  trivial  case  k=l  where  they  are 
eqiud.  This  shows  that  in  the  observation  of  events  based  on 
the  P's  a  greater  precision  may  be  expected  than  for  a  series  of 
observations  based  on  the  p's.  The  method  of  just  perceptible 
differences  and  the  method  of  irregular  variations  make  use  of 
the  P's,  but  the  error  methods  use  the  p's,  so  that  a  number  of 
observations  made  by  the  method  of  right  and  wrong  cases  will 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  69 

give  a  smaller  precision  than  the  same  number  of  observations 
by  the  method  of  just  perceptible  differences  or  by  the  method 
of  irregular  variation.  These  methods  do  not  undertake  as  much 
as  the  method  of  right  and  wrong  cases  does,  but  what  they  do 
they  do  with  a  great  amount  of  precision. 

On  the  basis  of  these  considerations  we  may  give  the  follow- 
ing description  of  the  method  of  just  perceptible  differences. 
Prepare  a  series  of  comparison  stimuli  which  cover  an  interval 
in  the  beginning  of  which  the  subject  gives  a  very  high  percent- 
age of  "smaller "-judgments  and  at  the  end  of  which  there  exists 
a  similar  high  probability  for  "greater "-judgments.  Be  careful 
to  apply  the  standard  stimulus  and  the  comparison  stimulus 
always  in  the  same  order  so  as  to  keep  the  time  error  con- 
stant. Present  the  pairs  in  any  order  and  record  all  the  judg- 
ments given.  If  the  investigation  is  somewhat  extended  it  is 
necessary  to  exhaust  the  possible  orders  systematically,  so  as 
not  to  favor  any  single  one.  Use  different  arrangements  of  com- 
parison stimuli.  All  the  stimuli  of  the  same  arrangement  must 
be  gone  through,  before  the  same  stimulus  is  presented  to  the 
subject  another  time.  Find  from  the  records  the  smallest  com- 
parison stimulus  in  every  series  on  which  the  judgment  "great- 
er" was  given,  and  the  largest  stimulus  on  which  another  but 
a  "greater "-judgment  was  given.  A  comiDarison  stimulus  is  the 
smallest  in  a  series  to  be  judged  "greater",  if  all  the  compari- 
son stimuli  of  smaller  intensity  were  judged  "smaller"  or 
"equal"  and  if  this  stimulus  was  judged  "greater."  A  compari- 
son stimulus  is  the  largest  on  which  a  "greater "-judgment  was 
not  given,  if  on  this  stimulus  either  one  of  the  judgments 
"smaller"  or  "equal"  is  passed  and  if  all  the  larger  comparison 
stimuli  were  judged  "greater".  These  are  the  data  for  the 
determination  of  the  just  perceptible  and  of  the  just  impercep- 
tible positive  difference.  Comljine  the  results  of  the  different 
series  by  taking  the  arithmetical  mean  of  all  the  ol)servations 
for  the  determination  of  the  just  perceptible  positive  difference 
and  for  that  of  the  just  imperceptible  positive  difference.  The 
average  of  the  just  perceptible  and  of  the  just  imperceptible 
positive  difference  gives  that  amount  of  difference,  for  which  there 
exists  the    probability  h  that    the  judgment    "greater"    will  be 


70  PROBLEMS  OF  PSYCHOPHYSICS 

given.  This  value  is  called  the  threshold  in  the  direction  of  in- 
crease. Find  from  the  same  records  the  largest  stimulus  on 
which  the  judgment  "smaller"  was  given,  and  the  smallest 
stimulus  on  which  a  "smaller "-judgment  was  not  given.  A 
stimulus  is  the  largest  of  a  series  on  which  the  judgment 
"smaller"  was  given,  if  this  stimulus  was  judged  smaller  and  if  all 
the  larger  differences  were,  judged  "equal"  or  "greater".  A 
stimulus  is  the  smallest  on  which  the  judgment  "smaller" 
was  not  given,  if  this  stimulus  was  judged  "equal"  or  "greater" 
and  if  all  the  smaller  stimuli  were  judged  "smaller".  Find  the 
just  perceptible  and  the  just  imperceptible  negative  difference 
by  taking  the  averages  of  all  the  observations  and  combine  the 
results.  The  final  result  is  that  amount  of  difference  for  which 
there  exists  the  probability  ^  that  the  judgment  "stnaller"  will 
be  given.  This  quantity  is  called  the  threshold  in  the  direction 
of  decrease. 

The  result  of  the  so-called  method  of  just  perceptible  differ- 
ences has,  therefore,  a  signification  which  does  not  depend  on 
the  notion  of  a  difference  which  is  always  perceived.  The  result 
of  this  method  is  expressible  in  the  fundamental  terms  of 
the  error  methods  and  it  was  seen  that  this  method  has  several 
decisive  advantages  over  the  other  methods  Avhich  use  the  same 
type  of  experiments  for  the  measurement  of  sensitivity. 
The  method  of  just  perceptible  differences  has  the  feature  of  an 
eminenth^  practical  method,  in  so  far  as  the  experiments  required 
are  very  simple,  as  its  algorithm  is  extremely  easy  and  its  degree 
of  accuracy  is  great,  so  that  only  a  relatively  small  number  of 
observations  is  required.  For  all  the  ordinary  purposes  of  a  de- 
termination of  the  thi;eshold  one  may  use  the  method  in  the 
form  described  by  the  previous  investigators  and  one  will  obtain 
serviceable  results.  For  all  purposes,  however,  which  require 
some  accuracy  one  will  not  fail  to  keep  a  complete  record  of  all 
the  judgments  given.  It  will  be  seen  later  that  such  a  set  of  re- 
sults can  be  treated  rnore  exhaustively  by  another  method,  which 
combines  the  advantages  of  the  error  methods  with  those  of 
the  method  of  just  perceptible  differences  and  which  yields  the 
result  of  the  latter  method  almost  without  any  work. 

This  description  of  the  method  of  just  perceptible  differences 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  71 

may  be  illustrated  by  the  example  of  our  experiments  on  lifted 
weights.  The  judgments  "greater"  and  "smaller"  correspond 
TO  our  judgments  ''heavier"  and  "lighter",  and  the  "guess"-judg- 
ments  of  our  experiments  are  equivalent  to  the  equality  cas.s. 
From  the  observed  frequencies  of  these  three  types  of  judgments 
we  may  find  their  probabilities  for  the  differences  which  were 
used.  The  theorem  of  Bernoulli  gives  the  limits  of  accuracy 
of  these  determinations.  These  probabilities  are  the  p's  in  our 
formulae  and  we  may  find  from  them  the  "value  of  the  P's  for 
every  difference  used  in  the  experiments  by  formula  II.  The 
first  business  is  to  arrange  the  results  in  the  way  shown  in  Table 
2S  (Appendix  p.  186).  Each  one  of  our  seven  subjects  is  given 
a  double  column,  the  first  column  of  which  bears  the  heading 
"h"  and  the  second  the  heading  "not-h."  The  column  "h" 
gives  the  observed  relative  frequencies  of  "heavier "-judgments 
for  every  difference,  and  the  second  column  gives  the  difference 
of  this  number  from  the  unit,  i.  e.  the  relative  frequencies  of 
judgment?  which  are  not  "heavier "-judgments.  From  these 
numbers  one  finds  the  values  of  the  P's,  beginning  at  the  top 
of  the  table,  successively;  these  values  are  given  in  Table  29 
(Appendix  p.  186).  This  computation  does  not  take  much  time 
if  one  considers  that  the  product  in  formula  II  contains  the  q's 
for  all  the  preceding  stimuli.  One  divides  conveniently  the  com- 
putation in  two  steps,  the  first  of  which  consists  in  finding  the 
sum  of  the  logarithms  of  the  q's  and  the  second  in  adding  the 
logarithm  of  the  corresponding  pj..  A  glance  at  the  results  in 
Table  29  shows  that  the  numbers  increase  at  first  and  decrease 
after  having  attained  a  maximum.  The  rise  and  fall  of  the 
numbers  is  uniform  not  being  interrupted  by  secondary  maxima 
except  for  subject  V  where  there  is  a  slight  notch  at  88.  The 
absence  of  interruptions  is  doubtlessly  due  to  the  fact  that  the 
intervals  between  the  comparison  stimuli  used  in  the  experi- 
ments are  rather  large.  It  will  be  noticed  furthermore,  that 
the  maximum  of  the  P's  coincides  with  one  of  the  stimuli  at  the 
beginning  or  at  the  end  of  the  interval  in  which  the  value 
p  =  ^  is  found.  It  so  happens  that  the  maximum  is  at  the  end 
of  the  interval  for  the  subjects  I  to  VI,  and  that  the  maximum 
i>  found  at   the  beginning  of  the  interval  only  for  the    subject 


72  PROBLEMS  OF  PSYCHOPHYSICS 

VII.  The  sum  of  all  the  P's  is  given  at  the  bottom  of  the  table. 
These  sums  are  slightly  different  from  the  unit,  which  indicate 
that  there  is  a  probability  which  is  not  negligible  that  the 
entire  series  may  be  presented  to  the  subject  without  giving 
a  result  for  the  determination  of  the  threshold.  This  circum- 
stance affords  a  very  convenient  way  of  checking  the  result 
of  the  computation.  A  series  of  experiments  will  not  give  a 
result  for  the  determination  of  the  just  perceptible  positive 
difference,  if  no  "heavier "-judgment  is  given  on  any  one  of  the 
stimuli.  The  probability  of  this  event  is  identical  with  the  pro- 
bability that  on  all  the  stimuli  judgments  are  given  which  are 
not  "heavier "-judgments,  and  this  probability  is  equal  to 
the  product  of  all  the  q's.  For  the  computation  of  the  last  P 
one  has  to  form  the  product  of  all  the  q's  with  the  exception  of 
the  q  for  the  last  stimulus.  The  product  of  all  the  q's,  there- 
fore, may  be  found  by  one  single  multiplication  and  the  result 
must  add  up  to  one  with  the  sum  of  all  the  P's.  The  products 
of  the  q's  are  given  in  the  line  marked  R.  The  sum  of  ^  and  R 
is  equal  to  the  unit  except  in  those  cases  where  it  was  necessary 
to  correct  the  individual  values  of  the  P's  in  such  a  way  that 
their  sum  involved  an  error  of  a  unit  at  the  last  decimal  place. 
We  may  make  the  remark  that  it  is  necessary  to  start  out  with 
four  correct  decimals  in  order  to  get  the  final  results  correct 
within  two  decimals. 

The  next  step  is  to  find  the  values  of  r^P;.and  of  rj^'Pj^,  which 
also  may  be  done  conveniently  by  successive  multiplication. 
Table  30  (Appendix  p.  187)  gives  the  value  of  rj^Pj^.  The  num- 
bers of  this  table  show  that  the  significant  values  are  clustered 
around  a  certain  point,  which  is  different  for  the  different  sub- 
jects, and  that  the  values  of  these  products  decrease  very  rap- 
idly on  both  sides  of  this  point.  This  indicates  that  the  influ- 
ence of  adding  new  stimuli  at  the  ends  of  the  series  is  inconsid- 
erable. It  seems,  however,  that  our  series  is  not  extended  enough 
for  a  determination  of  the  just  perceptible  positive  difference,  be- 
cause the  values  at  the  bottom  of  the  table  are  considerably  greater 
than  those  at  the  upper  end  of  the  table.  We  conclude  that  it 
would  be  advisable  in  a  series  of  experiments  which  has  the  main  ' 
purpose  of  ascertaining  the  threshold  of    difference    to  add  at 


METHOD    OV    JUST    PERCEPTIHLE    DIFFERENCES  73 

least  one  more  stimulus  at  the  upper  end  of  our  series  of  stimuli. 
This  is  also  suggested  by  the  fact  that  the  values  of  R  in  Table 
29  are  by  no  means  inconsiderable,  so  that  the  chances  are  that 
one  will  not  only  have  a  relatively  great  number  of  resultless  ex- 
periments, but  one  also  will  neglect  some  terms  which  are  of  influ- 
ence for  the  determination  of  the  threshold  of  difference.  The 
value  of  rj^Pj^  for  r=108  gives  an  indication  of  the  possible  influ- 
ence of  the  next  term  and  the  following  comparison  shows  that 
the  values  of  R  and  influence  of  the  last  term  go  parallel  to  some 
extent. 


Subject 

Value  of  R 

r  P  for  108  gr 

VII 

0.0189 

9.2772 

VI 

0.0128 

10.4976 

II 

0.0050 

5.0868 

I 

0.0025 

4.3092 

III 

0.0017 

4.1148 

IV 

0.0017 

2.4732 

V 

0.0009 

1.6524 

It  is  advisable  to  have  in  every  series  at  least  two  stimuli  for 
which  the  probability  of  a  "heavier"-judgment  is  very  close 
to  the  unit  because  the  product  of  the  q's  becomes  very  small  in 
this  case.  A  further  observation  which  is  impressed  by  Table 
30  is  the  marked  asymmetry  in  the  distribution  of  the  values. 
With  the  exception  of  subject  V  the  values  of  i\  F^  increase 
slowly  and  show  a  very  steep  descent,  whereas  in  the  subject  V 
the  course  is  opposite.  The  skewness  is  quite  unmistakable 
also  in  this  case,  but  the  slope  in  the  ascent  and  descent  of  the 
values  is  not  as  steep  for  this  subject  as  the  descent  of  the 
values  for  the  other  subjects. 

The  sum  of  all  the  r^  Pj,'s  is  given  at  the  bottom  of  Table  30. 
This  value  is  the  just  perceptible  positive  difference  for  this  series. 
If  we  were  dealing  with  symmetrical  distributions  this  value 
would  give  the  amount  of  difference  for  which  there  exists  the 
probability  ^  that  it  will  l^e  judged  "greater",  but  independently 
from  a  supposition  about  tlio  form  of  the  distrifjution  this  number 
has  the  signification  of  the  mathematical  expectation.  It  is 
noteworthy  that  all  these  valu3S  are  smaller  than  100  gr,  which 


74        '  PROBLEMS  OF  PSYCHOPHYSICS 

indicates  that  in  the  comparison  of  a  100  gr  standard  with  a  100 
gr  comparison  weight  a  number  of  "heavier "-judgments  will 
be  given  which  exceeds  50%.  The  numbers  in  our  table  vary- 
between  96.9139  and  99.8164,  and  it  is,  perhaps,  not  a  chance 
incident  that  the  smallest  value  is  that  of  the  female  observer. 
Her  value  is  by  more  than  1  gr  smaller  than  the  lowest  value  of  the 
other  subjects.  From  the  values  of  v^  P^  given  in  Table  30  we 
find  the  values  of  Tj.^  Pj^  immediately.  These  numbers  and 
their  sums  are  given  in  Table  31  (Appendix  p.,  187).  The  sum 
of  these  numbers  gives  the  mean  value  of  the  squares,  a  number 
which  is  needed  for  the  application  of  the  theorem  of  Tchebitcheff . 
Subtracting  from  the  sums  in  this  table  the  square  of  the  numbers 
at  the  bottom  of  the  preceding  table  and  taking  the  square  root 
from  the  result  gives  the  coefficient  which  we  need  for  the  com- 
putation of  the  probability  of  a  deviation  by  the  theorem  of  Tcheb- 
itcheff. We  obtain  in  this  way  the  numbers  15.11,  8.64,  5.97, 
6.23,  5.20,  12.30  and  14.41  for  our  seven  subjects  I.  II,....  VII. 
The  most  probable  result  for  the  determination  of  the  just  per- 
ceptible positive  difference  and  the  probabilities  for  deviations 
from  it  may  be  represented  in  this  way:     There  exists  a  proba- 

t^ 
bility  greater  than  1-  —  that  the  observed  value  will  fall   with- 
n 

in  the  limits 

15.11  15.11    ,       , 

99.76 and   99.76 -H for  the  subject       I; 

t  t 


8.64        ^  8.64      ^^     ^, 

98.31 "     98.31+-  "    "        "  II; 


^  97  5  97 

99.82--  "     99.82+-  "     "        "         III; 

t  t 


97.98  __^:^      -     97.98+   --''-     -     "         "  IV; 

t  t 


METHOD    OF   JUST    PERCEPTIBLE    DIFFERENCES 


75 


5  20  5  ''O 

96.91 ^       and   96.91+       '"     for  the  subject      V; 

t  t  ^ 


99.21  - 


12.30 


99.21  + 


12.30 


VI; 


98.76  - 


14.41 


98.76  + 


14.41 


VII. 


These  numbers  may  be  used  in  two  ways;  either  one  may  ask 
how  many  experiments  are  needed  in  order  to  find  the  result  in- 
side of  a  certain  interval  with  a  given  probability,  or  one  may  ask 
what  is  the  probability  that  a  given  deviation  will  be  observed 
in  a  certain  number  of  experiments.  This  affords  the  possibility 
of  comparing  the  theoretical  results  with  the  actual  observa- 
tions. We  may  ask,  for  instance,  what  is  the  interval  inside 
of  which  we  may  expect  the  result  with  the  probability  ^,  if 
for  each  of  the  subjects  I,  II  and  III  400  and  for  every  other 
subject  300  observations  on  the  just  perceptible  positive  difference 
were  made.  For  the  first  three  subjects  we  find  t  =  \/200  =  14.14 
and  for  the  other  subjects  t  =  \/l75  =  13.23.  The  observed  diff- 
erences from  the  theoretical  value  may  be  found  from  Table  39 
(Appendix  p.  191),  and  they  may  be  compared  with  the  results  of 
the  computation.  The  deviation  from  the  most  probable  result 
inside  of  which  the  observed  result  may  be  expected  with  a 
probability  greater  than  ^  is 


for  the       I  subject  1.07,  the  observed  difference  is  0.60 


"   "      II   " 

0.61      " 

"    0.40 

"   "     III   " 

0.42      " 

"    0.32 

"       u            jy       « 

0.47      " 

"    0.54 

"     "    •       V     " 

0.55      " 

"    0-78 

«       „               Y       « 

0.55      " 

"    0.78 

"     "        VI     " 

0.91      " 

"     1.20 

"     "      VII     " 

1.09      " 

"    2.11 

76  PROBLEMS  OF  PSYCHOPHYSICS 

Four  of  the  observed  differences  are  greater  and  three  are 
smaller  than  the  theoretical  result.  We  would  have  to  expect  such 
a  result  for  an  event  the  probability  of  which  is  approximately  \. 
It  is  noteworthy  that  the  observed  and  the  theoretical  deviations 
are  of  the  same  order  of  magnitude. 

In  order  to  obtain  the  empirical  determination  of  the  just  per- 
ceptible positive  difference  we  have  to  go  back  to  the  records  of 
the  experiments,  and  we  have  to  find  in  every  column  that  weight 
which  was  the  smallest  on  which  a  "heavier "-judgment  was 
given.  In  the  example  of  a  record  sheet  which  we  have  given  on 
page  6  we  find  in  the  first  column  92,  in  the  second  column  96, 
in  the  third  column  96,  in  the  fourth  column  100  and  in  the  fifth 
column  96  as  the  smallest  weight  on  which  the  judgment  "heavier" 
is  given.  In  this  way  one  may  find  for  each  comparison  weight, 
how  many  times  it  happened  that  it  was  judged  "heavier"  when 
all  the  smaller  weights  were  judged  "lighter"  or  "equal".  These 
results  are  given  in  the  Tables  32-38  (Appendix  pp.  188-191). 
Each  subject  is  given  a  separate  table,  which  contains  the  results 
of  every  series  of  100  experiments  in  a  double  column.  We  have, 
therefore,  400  determinations  of  the  just  perceptible  positive  dif- 
ference for  the  subjects  I,  II,  III  and  300  determinations  for  each 
of  the  other  subjects.  The  numbers  in  the  columns  under  the  head- 
ing Njj  give  for  every  weight  the  number  of  times  it  was  obtained 
as  a  determination  of  the  just  perceptible  positive  difference, 
and  the  numbers  opposite  in  the  next  column  give  the  product 
of  the  weight  multiplied  by  the  number  of  times  it  was  observed 
as  the  just  perceptible  difference  (rj^  Nj^).  This  is  the  number 
which  must  be  taken  for  every  weight  according  to  the  al- 
gorithm of  the  method  of  just  perceptible  differences.  The 
final  determination  is  obtained  by  dividing  the  sum  of  the  rj^  Njj 
by  the  total  number  of  observations,  i.  e.  by  the  sum  of  the  Nj^. 
The  sums  are  given  at  the  bottom  of  the  tables  in  the  line  marked 
^  and  the  results  of  the  final  determination  of  the  just  percep- 
tible positive  difference  are  given  in  the  line  marked  "Average". 
It  will  be  remarked  that  the  sum  of  the  Nj.  is  not  always  equal 
to  100.  This  is  a  consequence  of  the  fact  that  the  sum  of  the  Pj^ 
is  not  exactly  equal  to  the  unit.  The  number  of  those  cases 
which  did  not  give  a  result  for  the  determination  of  the  just  per- 


METHOD    OV    JUST    PERCEPTIBLK    DIFFERENCES  I  I 

ceptible  positive  difference  is  small,  as  it  may  be  expected  since 
the  smallness  of  the  differences  of  the  sums  of  the  P's  from  the 
vmit  indicates,  that  it  is  a  rather  rare  event  that  all  the  pairs 
are  presented  to  the  subject  without  a  single  weight  being  judged 
"heavier".  This  event  happened  most  frequently  with  the  sub- 
jects II,  VI  and  VII,  for  whom  the  difference  of  the  sum  of  the 
P's  from  the  unit  is  largest.  The  comparison  of  the  observed  results 
for  the  determination  of  the  just  perceptible  difference  with  the 
calculated  results  given  in  Table  30  shows  that  the  coincidence 
of  the  observed  results  with  the  theoretical  results  is  very  satisfac- 
tory even  with  a  relatively  small  number  of  observations.  An 
analysis  of  this  coincidence  is  given  in  Table  39  (Appendix  p.  191). 
where  it  is  shown  how  much  the  result  of  the  observations  differs 
from  the  theoretical  result  in  every  series  of  100  experiments. 
The  deviations  are  very  small  and  they  comply  with  the  test  of 
Tchebitcheff's  theorem  in  a  very  high  degree.  In  a  small  ma- 
jority of  cases  the  observed  result  is  found  inside  the  interval 
for  which  there  exists  a  probability  greater  than  h.  The  last 
column  of  this  table  gives  the  combined  result  of  all  the  observa- 
tions. The  numbers  are  found  from  the  data  of  Table  40  (Ap- 
pendix p.  192),  which  shows  the  results  of  the  determination  of 
the  just  perceptible  difference  if  the  results  of  all  the  series  are 
combined.  The  signification  of  Nj^  and  rj^Ni,  is  the  same  as  in 
the  previous  tables.  Every  subject  is  given  a  double  column 
and  the  numbers  Nj^  in  this  table  are  equal  to  the  sum  of  the  cor- 
responding values  given  above  in  the  table  for  the  same  subject. 
A  test  of  the  correctness  of  the  computation  consists  in  compar- 
ing the  sum  of  rj^Nj,  which  is  equal  to  the  sum  of  the  correspond- 
ing values  in  the  former  table  for  the  same  subject.  In  close 
agreement  with  the  results  which  were  found  above  we  find  in 
the  majority  of  cases  the  value  of  the  just  perceptible  positive 
difference  below  100,  and  in  those  cases  where  it  exceeds  100  the 
difference  is  very  small.  The  term  "just  perceptible  positive 
difference ' '  is,  therefore,  somewhat  misleading,  because  the  actual 
difference  between  the  two  stimuli  will  be  negative  in  most  cases. 
This  is  due  to  the  fact  that  the  time  error  was  not  eliminated  in 
our  experiments.  One  must  keep  in  mind  that  we  use  the  term 
"just  perceptible  positive  difference"  merely  as  an  abbreviation 


78  PROBLEMS  OF  PSYCHOPHYSICS 

for  the  comparipon  stimulus  of  the  pair,  which  is  obtained  as  a 
result  of  a  determination  of  the  just  perceptible  positive  difference 
by  the  method  of  just  perceptible  differences.  We  know  that 
there  exists  for  this  stimulus  the  probability  h  that  the  com- 
parison with  the  standard  will  result  in  the  judgment  "greater". 
It  is  not  to  be  feared  that  a  serious  misunderstanding  may  arise 
from  this  terminology,  and  for  this  reason  it  does  not  seem  ad- 
visable to  change  a  terminology  which  is  more  or  less  commonly 
in  use.  The  negative  sign  of  this  difference  indicates  that  one  must 
not  start  from  objective  equality  of  the  stimuli  when  applying  the 
method  of  just  perceptible  differences  but  from  subjective  equal- 
ity, a  fact  which  was  recognised  alsa  by  previous  investigators 
of  this  method.  The  judgment  which  is  passed  on  the  compari- 
son of  the  stimuli  depends  on  a  number  of  conditions  among 
which  the  actual  difference  of  the  stimuli  is  only  one.  We  have 
to  distinguish  between  the  actual  difference  and  the  effective 
difference,  to  use  a  term  of  G.  E.  Miiller,  understanding  by  this 
term  the  difference  of  a  stimulus  from,  subjective  equality. 

The  next  step  to  be  taken  in  working  out  the  data  by  the  method 
of  just  perceptible  differences  is  the  computation  of  the  just  im- 
perceptible positive  difference.  The  necessary  data  for  this  com- 
putation are  given  in  Table  28.  Beginning  at  the  bottom  of 
the  table  we  have  to  form  successively  the  products  of  the  p's  and 
multiply  them  by  the  corresponding  values  of  c^.  In  this  way  one 
finds  the  numbers  given  in  Table  41  (Appendix  p.  192),  which 
are  the  P's  for  the  computation  of  the  just  imperceptible  posi- 
tive difference.  The  P's  for  108  are,  of  course,  identical  with  the 
probability  that  on  this  weight  a  judgment  will  be  given  which 
is  not  a  "heavier"-judgment.  The  course  of  the  numbers  in 
this  table  is  similar  to  that  of  the  numbers  in  Table  29.  It  is 
perfectly  regular,  not  a  single  break  in  the  ascent  or  descent  oc- 
curring throughout  the  entire  table.  In  some  cases  the  ascent  is 
very  rapid  the  maximum  being  reached  almost  with  a  jump 
(subject  I,  III  and  IV),  in  other  cases  the  ascent  and  descent 
take  place  with  approximately  equal  rapidity  (e.  g.  subject  V), 
so  that  without  further  investigation  one  almost  might  speak  of 
a  symmetrical  distribution.  The  values  of  the  P's  become  very 
small  for  88  and  84,  and  since  the  values  for  108  are  rather  large, 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  79 

it  makes  the  impression  as  if  the  table  of  the  distribution  of  the 
P's  were  cut  off  at  the  upper  end.  A  consequence  of  the  small 
values  of  P  for  the  last  differences  of  the  table  is  that  the  values 
of  R  are  insignificant  for  all  the  seven  subjects.  In  no  case  has 
R  a  counting  figure  on  the  fourth  decimal  place.  The  sum  of 
the  P's  may  differ  by  the  unit  of  the  last  decimal  place  from  one 
owing  to  necessary  corrections  of  the  single  P's. 

Multiplying  the  numbers  of  Table  41  with  the  corresponding 
r's  one  finds  the  values  r,^  F^,  which  are  given  in  Table  42  (Ap- 
pendix p.  193).  It  becomes  evident  from  this  table  that  the 
series  of  comparison  stimuli  is  extended  enough  in  the  direction 
of  decrease,  but  that  it  is  too  short  in  the  upper  part.  The  lower 
part  of  this  table  shows  very  well  how  rapidly  the  influence  of 
a  comparison  stimulus  becomes  insignificant,  if  it  is  somewhat  dis- 
tant from  the  most  probable  result.  The  sum  of  all  the  terms 
^k  ^k  gives  the  just  imperceptible  positive  difference,  which  is 
given  at  the  bottom  of  Table  42  in  the  line  marked^.  It  will 
not  be  void  of  interest  to  compare  the  theoretical  values  of  the 
just  perceptible  and  of  the  just  imperceptible  positive  differences. 


1 1^  1  p  p  t 

Just  perceptible 

Just  imperceptible 

LUJCL'  I, 

positive  difference. 

positive  difference 

I 

99.7642 

99.1388 

II 

98.3050 

99.3640 

III 

99.8164 

98.7522 

IV 

97.9791 

98.1866 

\^ 

96.9139 

97.3702 

\T 

99.2082 

100.5176 

VTI 

98.7590 

100.9592  . 

These  numbers  show  that  the  just  imperceptible  positive 
difference  is  smaller  than  the  just  perceptil^le  positive  difference 
in  the  case  of  two  subjects  (I  and  III)  and  that  it  is  larger  in 
all  the  other  cases.  Several  investigators  have  made  the  obser- 
vation that,  with  their  subjects,  the  just  imperceptible  positive 
differences  had  the  tendency  to  be  smaller  than  the  just  percep- 
tible positive  differences.  This  fact  was  attributed  to  the  influ- 
ence of  attention  and  it  seemed  to  be  an  objection  against  the 
method  of  just  perceptible  differences.    Our  results  bear  out,  what 


80  '  PROBLEMS  OF  PSYCHOPHYSICS 

one  easily  can  verify  by  theoretical  considerations,  that  in  some 
cases  the  just  perceptible  positive  difference  will  be  fj;reater  than 
the  just  imperceptible  difference  and  that  in  other  cases  the  oppo- 
site will  take  place.  It  depends  on  the  chance  influence  of  the 
choice  of  the  comparison  stimuli  which  difference  will  be  greater. 
Multiplying  the  values  r^Pj^  by  r^  gives  the  values  v^'P^^  which 
are  given  in  Table  43  (Appendix  p.  193).  The  mean  value  of 
the  squares  is  needed  for  the  determination  of  the  limits  inside 
of  which  the  result  may  be  expected  to  fall  with  a  probability 

t^ 
exceeding  1- —  .     These  intervals  are 


3.86  3.86 

99.14 and  99.14  +  ^ for  the  subject  I; 

t  t 


4.58                          4.58 
99.36 "    99.36+ "     "        "      II; 


•?  71                           3  71 
98.75 -_lii-    "    98.75  + "     "        "    III: 


98:19-  ^'^^      "    98.19  +  -^^^^    "     "        "      IV; 


t 

4.64 

t 

5.10 

t 

4.24 

5  10 
97.37-——-    "     97.51  +  ^^^^    "     "        "        V; 


4.24  

100.52 "100.52  + "     "        "      VI; 


4  5'^  4  52 

100.96 '-^-    "100.96+-^ "    "        "     VII. 

t  t 

The  coefficients  which  determine  the  size  of  the  interval  are 
smaller  for  the  just  imperceptible  positive  difference  than  for 
the  just  perceptible  positive  difference,  and  we  may  expect  that 


METHOD    OF   JUST    PERCEPTIBLE    DIFFERENCES  81 

the  coincidence  of  the  observed  results  with  the  theoretical  results 
is  even  closer  than  it  was  in  the  first  case. 

For  the  empirical  determination  of  the  just  imperceptible  posi- 
tive difference  we  have  to  find  from  the  records  how  many  times 
every  weight  was  the  largest  on  which  a  judgment  was  given 
which  was  not  a  "heavier  "-judgment.  In  the  example  of  a  record 
sheet  on  p.  6  we  have  100  in  the  first  column,  92  in  the  second 
column,  92  in  the  third  column,  96  in  the  fourth  column  and  92 
in  the  last  column  as  empirical  deterininations  of  the  just  imper- 
ceptible positive  difference.  Counting  the  records  over  in  this 
way  one  obtains  the  data  for  the  construction  of  the  Tables  44-59 
(Appendix  pp.  194-197),  which  are  similar  in  their  construction 
to  the  Tables  32-38.  Each  subject  is  given  a  separate  table, 
which  contains  in  every  double  column  the  results  of  a  series  of 
100  experiments.  The  numbers  in  the  column  Nj^  refer  to  the 
number  of  times  a  weight  was  obtained  as  a  determination  of  the 
just  imperceptible  positive  difference.  The  numbers  in  the  col- 
umns rj^Nj.  give  the  product  of  the  weight  with  the  number  of 
times  it  was  obtained  as  a  determination  of  the  just  imperceptible 
positive  difference.  The  numbers  at  the  bottom  of  the  tables 
give  the  sums  of  all  the  numbers  in  the  corresponding  columns. 
The  averages  of  the  numbers  r,^  Nj^  are  the  empirical  determina- 
tions of  the  just  imperceptible  positive  difference  in  these  partic- 
ular series.  The  results  of  the  observations  agree  very  well  not 
only  with  the  theoretical  results  but  also  with  the  results  of  the 
empirical  determinations  of  the  just  perceptible  positive  differ- 
ence. We  have  seen  that  in  the  case  of  a  symmetrical  distril)u- 
tion  the  just  perceptible  positive  difference  is  that  amount  of 
difference  for  which  there  exists  the  probability  \  that  a  "heav- 
ier "-judgment  will  be  given,  and  that  under  the  same  condition 
of  a  symmetrical  distribution  the  just  imperceptible  positive  differ- 
ence is  that  amount  of  difference  for  which  there  exists  the 
probability  \  that  a  "heavier "-judgment  will  not  be  given. 
This  is  of  course  the  same  quantity  in  both  cases  and  empir- 
ical determinations  of  the  just  perceptible  positive  difference  and 
of  the  just  imperceptiljle  positive  difference  must  give  results 
which  are  sensibly  equal.  The  comparison  of  these  numbers  is 
given  in  Table  51   (Appendix  p.  197),  which    contains    in    every 


82  PROBLEMS  OF  PSYCHOPHYSICS 

double  column  the  results  for  the  just  perceptible  and  for  the 
just  imperceptible  positive  difference  of  one  series  of  100  experi- 
ments. The  columns  under  the  headinf^  A  give  the  results  of  the 
determination  of  the  just  perceptible  positive  difference,  and  those 
under  the  heading  B  give  the  just  imperceptible  positive  differ- 
ence. The  largest  difference  between  the  corresponding  numbers 
in  the  columns  A  and  B  is  2.92  gr  (subject  III,  series  I),  and  the 
smallest  is  0.04  gr  for  the  same  subject  in  the  series  IVa  and 
in  the  series  IV.  The  differences  between  these  values  are  not 
large,  and  small  differences,  furthermore,  interchange  with  large 
differences;  it  is,  however,  remarkable  that  the  just  perceptible 
difference  is  greater  than  the  just  imperceptil)le  difference  in  a 
large  majority  of  the  cases. 

The  results  of  all  the  series  of  100  experiments  each  may  be 
combined  and  a  determination  of  the  just  imperceptible  difference 
may  be  oljtfiined  from  all  the  data.  The  combined  results  are 
given  and  worked  out  in  Talile  52  (Appendix  p.  198),  which  is 
constructed  in  the  same  way  as  Table  40  giving  the  results  for 
each  one  of  the  7  subjects  in  one  double  column.  It  is  of  interest 
to  notice  that  the  sum  of  the  N's  differs  from  the  total  number 
of  experiments  for  two  subjects  (IV  and  VII).  This  indicates 
that  it  is  rather  rare  that  the  series  is  gone  through  without 
one  weight  bei;ig  judged  "not-h."  This  agrees  with  Tal)le  41  al- 
though the  observed  values  are  somewhat  larger  than  we  may 
expect  on  the  basis  of  the  numbers  of  this  table.  The  results 
of  the  final  determination  of  the  just  imperceptible  positive 
difference  compare  in  this  way  with  those  for  the  just  percep- 
tible positive  difference. 

„   ,  .  Just  perceptible         Just  imperceptiljle  .^ 

oUDjeci.  .  .        Tff  •,•        !•«■  JJirierence. 

positive  difference,      positive  difference. 

I  100.36  98.84     .  +1.52 

II  98.71  98.71  0.00 

HI  99.50  99.67  -0.17 

IV  98.52  97.97  -fO.55 

V  97.69  97.00  +0.69 

VI  100.41  100.25  +0.16 

VII  100.87  98.37  +2.50  . 


METHOD    OF    JUST    I'EUCKPTIH  LK    DIFFEKKXCES  83 

A  Striking  feature  of  this  comparison  is  tlic  sign  of  the  diff- 
erences. The  just  perceptible  positive  difference  is  almost  invari- 
ably larger  than  the  just  imperceptible  positive  difference.  This 
observation  agrees  with  the  results  of  previous  investigators  of  the 
method  of  just  perceptible  differences,  who  as  a  rule  attributed 
this  fact  to  the  particular  order  in  which  the  comparison  stimuli 
were  presented.  It  was  supposed  that  a  small  stimulus  was 
more  readily  perceived,  if  it  was  preceded  b}^  several  other  small 
stimuli  of  decreasing  intensity,  because  attention  was  better 
adapted  by  the  preceding  stimulation.  This  explanation  can 
not  hold  for  our  experiments  because  the  stimuli  were  presented 
in  very  different  order  and  an  adaptation  of  attention,  there- 
fore, cannot    take  place. 

We  now  turn  to  the  description  of  our  ol)servati()ns  on  the 
just  perceptible  and  on  the  just  imperceptible  negative  difference. 
The  empirical  determinations  of  the  probabilities  of  a  "  lighter  "- 
judgment  are  the  data  which  we  need  for  this  computation. 
These  numbers  are  given  in  Table  53  (Appendix  p.  198)  for 
each  subject  in  one  double  column.  The  numbers  in  the  columns 
marked  "1"  give  the  probabilities  of  "lighter " -judgments 
for  the  various  differences.  The  numbers  in  the  columns  mark- 
ed "not-1"  give  the  probabilities  of  judgments  which  are  not 
''lighter "-judgments.  These  probabilities  are  of  course  equal 
to  the  differences  of  the  numbers  "1"  from  the  unit.  The  num- 
bers "1"  and  "not-l"  are  the  values  u  and  v  in  our  theoretical 
deductions  and  we  find  from  them  the  U's  by  successive  multipli- 
action  beginning  at  the  bottom  of  the  table.  The  values  of  U  and 
u  are  identical  for  the  comparison  weight  108  gr,  as  may  be  seen 
from  Table  54  (Appendix  p.  I99j,  where  the  values  of  the  U's 
are  given.  The  increase  and  decrease  of  the  U's  is  uniform 
throughout  the  whole  table.  The  sums  of  the  U's  are  very  close  to 
the  unit  as  may  be  seen  from  the  values  of  R,  which  are  given 
at  the  bottom  of  table.  It  is  instructive  to  compare  the  values 
of  R  in  this  table  with  those  of  Table  29. 


84  PROBLEMS  OF  PSYCHOPHYSICS 


Subject. 

Value 

of  R  in  Table  29. 

Value 

of  R  in  Table  54 

I 

0.0025 

0.0022  . 

II 

0.0050 

0.0010 

III 

0.0017 

0.0000 

IV 

0.0017 

0.0003 

V 

0.0009 

0.0005 

VI 

0.0128 

0.0004 

VII 

0.0189 

0.0002  . 

The  values  of  R  for  the  just  perceptible  positive  difference 
are  throughout  greater  than  those  for  the  just  perceptible  nega- 
tive difference,  and  only  for  the  first  subject  does  the  latter  value 
come  anywhere  near  the  first.  This  indicates  that  the  series  of 
comparison  weights  is  better  adapted  for  the  determination  of 
the  just  perceptible  negative  difference.     The  largest  value  of 

R  in  Table  54  is  0.0022  or  approximately  g^-  Since  R  gives 
the  probability  that  the  series  of  stimuli  will  be  presented  to  the 
subject  without  giving  a  result  for  the  method  of  just  percepti- 
ble differences  —  a  fact  which  one  will  try  to  avoid  for  obvious 
reasons  —  one  may  be  satisfied  with  this  series  of  stimuli  in  a  de- 
termination of  the  threshold  which  does  not  aim  at  a  higher  ac- 
curacy than  it  is  attainable  in  some  500  experiments.  For  the 
subjects  III  and  IV  one  even  could  leave  out  the  comparison 
stimlus  84  without  seriously  impairing  the  efficiency  of  the  series. 
This  observation  indicates  that  the  series  of  comparison  stimuli 
must  be  adapted  to  the  individual  whose  sensitivity  is  to  be 
tested,  and  that  superfluous  experimenting  may  be  avoided  by 
an  appropriate  choice  of  the  stimuli.  The  values  of  the  u's  serve 
for  the  computation  of  the  products  i\  V^  and  r^.-Uk^  which  are 
given  in  Tables  55  and  56  (Appendix  p.  199  sq).  which  are  needed 
for  the  calculation  of  the  just  perceptible  negative  difference  and 
for  the  determination  of  the  reliability  of  this  determination.  The 
data  of  these  two  tables  allow  to  represent  the  determination  of  the 
just  perceptible  negative  difference  in  the  following  way.  There 

t^ 
exists  a  probability  greater  than  1 that    the    result  of    an 

observation  of  this  quantity  will  fall  within  the  limits 


METHOD    OF   JUST    Pf:HCEPTIBLE    DIFFERENCES  85 


93.08  -  -^—  and  93.08  +  — —  for  the  subject      I; 
t  t 


7.89  7.89 

95.28 •'    95.28  + "     "         "        II; 

t  t 


97.40--^^    "  97.40  + -5:^    "     "  "       III; 

t  t 


5.74  5.74 

95.23--'' "95.23  + "      "  "       IV; 

t  t 


5.21  5.21 

94.20 "   94.20+       —    "      "  "         V; 

t  t 


95.25-^^    "    95.52  +  ^^^     "     "         "        VI; 


96.04-^^^    "'94.04  +  —-    "     "         "      VII. 
t  t 

These  values  are  found  in  the  same  way    as  the  corresponding 
values  previously  given,  namely  by  the  algorithm 

1 


I^Pzk-rV^'kPv-i^'kP^y. 


k 


The  signification  of  these  numbers  is  similar  to  that  of  the 
numbers  given  in  Tables  30  and  31,  so  that  it  does  not  need  be 
discussed  any  more.  We  proceed  immediately  to  give  the  re- 
sults of  the  observations  on  the  just  perceptible  negative  difference 
A  stimulus  will  be  recorded  as  an  observation  of  the  just  percep- 
tible negative  difference,  if  it  is  the  largest  on  which  the  judg- 
ment "lighter"  was  given,  all  the  larger  stimuli  being  judged 
"heavier"  or  "equal."     In  the  sample  of  a  record  sheet  given 


86  PROBLEMS  OF  PSYCHOPHYSICS 

on  p.  G  we  have  in  the  first  column  88,  in  the  second  84,  in  the 
third  column  92,  in  the  fourth  column  92  and  in  the  fifth  col- 
ums  88  as  determinations  of  the  just  perceptible  negative  dif- 
ference. These  results  of  our  experiments  are  given  in  the 
Tables  57-63  (Appendix  pp.  200-203),  which  are  constructed  in 
the  same  wa}'  as  the  corresponding  tables  for  the  just  perceptible 
positive  dffference.  The  results  for  each  subject  are  given  in 
a  separate  table,  which  contains  the  results  of  every  series  of 
100  experiments  in  one  double  column.  The  headings  N,^  and 
ri.Ni.  have  the  same  signification  as  before.  The  sums  of  Ni^  and 
those  of  the  products  Ti^  N^  are  given  at  the  bottom  of  every  col- 
umn and  the  final  result  of  the  determination  of  the  just  percep- 
tible negative  difference  is  given  in  the  line  marked  "Average". 
The  following  table  (Table  64,  Appendix  p.  204)  gives  the  deter- 
mination of  the  just  perceptible  negative  difference  if  all  the 
results  are  combined.  Our  theoretical  considerations  have  shown 
that  we  may  expect  the  results  to  fall  within  rather  narrow  limits. 
The  results  verify  this  expectation  in  so  far  as  our  observations 
show  not  only  a  close  coincidence  with  the  theoretical  results, 
but  the  results  of  the  different  series  of  100  experiments  of  the 
same  subject  also  show  very  little  variation  from  each  other. 
This  coincidence  is  illustrated  by  the  data  of  Table  65  (Appendix 
p.  205),  which  gives  under  the  heading  IVa,  I,  III,  IV  the  results 
for  the  different  series,  and  in  the  column  "Total"  the  result  of 
the  determination  of  the  just  perceptible  negative  difference  for 
the  combined  results.  If  the  coincidence  of  the  observed  results 
with  the  theoretical  results  is  great  in  the  series  of  100  experi- 
ments one  cannot  help  calling  this  coincidence  for  the  more 
extended  series  surprising.  The  maximum  of  the  deviations  is 
0.18  gr,  the  minimum  is  0.00  and  the  average  is  0.076  gr.  The 
number  of  the  positive  and  that  of  the  negative  deviations  are 
approximately  equal.  This  shows  that  the  method  of  just  per- 
ceptible differences  is  capable  of  great  exactitude  under  favorable 
circumstances. 

We  now  turn  to  the  study  of  the  just  imperceptible  negative 
difference,  the  data  for  the  computation  of  which  we  find  in  Table 
53  (Appendix  p.  198).  Applying  the  formula  based  on  the  de- 
finition of  the  just   imperceptible  negative  difference  one  finds 


MKTHOD    OF    .IIST    PERCEPTIBLE    DIFFEREXCES  87 

by  successive  multiplication,  beginning  at  the  top  of  Table  53, 
the  values  of  the  U's,  which  are  given  in  Table  66  (Appendix  p. 
205).  The  values  of  U  for  84  are  identical  with  the  probabilities 
of  a  "not-T'-judgment  for  this  weight.  The  sums  of  the  U's  are 
only  slightly  different  from  the  unit,  indicating  that  the  proba- 
bility is  very  small  that  a  series  will  end  resultless.  In  one  case 
(subject  I)  it  happened  that  the  sum  of  the  U's  exceeds  one  by 
the  unit  of  the  last  decimal  place,  -which  is  due  to  the  necessity 
of  correcting  the  single  values  of  the  U's.  The  course  of  the  in- 
crease and  decrease  of  the  U's  is  regular  throughout  the  entire 
table.  From  these  values  of  the  U's  one  finds  the  products 
rj^Ujj  and  Tj^^Uj^,  which  are  given  in  the  following  two'  tables(Tables 
67  and  68,  Appendix  p.  206).  The  sum  of  the  r^JJ^^  is  the  de- 
termination of  the  just  imperceptible  negative  difference.  These 
values  compare  thus  with  the  just  perceptible  negative  difference: 


Subject. 

Just  percept i 
negative  differ 

ble 
ence. 

Just  imperceptible 
negative  difference. 

I 

93.0849 

93.5089 

II 

95.2780 

94.4648 

III 

97.4041 

98.2963 

IV 

95.2281 

95.5572 

V 

94.1966 

94.7417 

VI 

95.5218 

95.0922 

VII 

96.0379 

95.5472  . 

The  just  perceptible  negative  difference  is  smaller  than  the 
just  imperceptible  difference  in  four  cases  and  larger  in  three 
cases.  There  is  in  our  results  no  indication  of  a  constant  rela- 
tion Ijetween  the  just  perceptible  and  the  just  imperceptible 
negative  difference.  The  data  of  the  last  two  tables  may  be  used 
for  representing  the  results  in  this  way.     There  exists  a  probabil- 

itv  greater  than  1 that  the  result  of  the  observation  will  fall 

n 

within  the  limits 


PROBLEMS  OV  PSYCHOPHYSICS 


4.46  ,  4.46    ,       ,         ,  . 

93.51 and  98.51  + for  the  subject         I; 

t  t 


4.71  4.71 

94.46 "     94.46+        -      "     "        "  ,11; 

t  t 


4  99  4  99 

98.30--'     -     "     88.30  +  —^-       "     "       "  HI; 

Xi  t 


5  67  5  67 

94.74  _J:^_      "     94.74+      '  "     "         "  IV; 

t  t 


4  19  4  19 

94.74-      ■  "    94.74+     '-      "     "         "  V; 

t  t 


4  93 
95.09 '  "     95.09  +  ^^^      "     "         "  VI; 

t 


4  78                           4.78 
95.55 "    95.55+ "     "        "         VII. 


The  coefficients  which  determine  the  limits  within  which  we  may 
expect  the  result  are  small,  so  that  a  close  coincidence  of  the 
theoretical  results  with  the  observed  results  may  be  expected. 
The  just  imperceptible  negative  difference  is  the  smallest  stim- 
ulus of  a  series  on  which  a  judgment  is  given  which  is  not  a  "light- 
er"-judgment.  In  the  example  of  a  record  on  p.  6,  we  have  in 
the  first  column  84,  in  the  second  column  88,  in  the  third  column 
96,  in  the  fourth  column  96  and  in  the  fifth  column  92  as  determi- 
nations of  the  just  imperceptible  negative  difference.  The  re- 
sults of  our  experiments  are  given  in  Tables  69-75  (Appendix  pp. 
207-209,211),  in  the  same  way  as  in  the  previous  tables  and  the  sig- 
nification of  the  letters  is  also  the  same.  The  next  table  (Table 
76,  Appendix  p.  210)  gives  the  combined  results  for  the  determi- 
nation of  the  just  imperceptible  negative  difference.     In  order 


METHOD  OF  JUST  PRECEPTIBLE  DIFFERENCES  89 

to  make  the  comparison  between  the  theoretical  and  the  observ- 
ed results  easier  Table  77  (Appendix  p.  211)  was  constructed 
which  shows  the  coincidence  of  these  quantities.  The  differences 
between  the  observed  and  the  theoretical  results  in  the  different 
series  of  100  experiments  are  small,  and  the  differences  between 
the  theoretical  results  and  the  results  of  the  combined  series  (see 
Ta])le  77  under  the  heading  "Totals")  are  so  inconsiderable 
that  they  ma}^  be  neglected  for  all  practical  purposes;  the  only- 
exception  is  subject  VII,  for  whom  this  difference  is  somewhat 
considerable.  This  is  due  to  the  fact  that  for  this  subject  all  the 
deviations  from  the  most  probable  result  are  positive.  This  large 
difference  between  the  observed  and  the  theoretical  values  for 
subject  VII  may  be  looked  at  as  just  as  much  of  a  chance 
as  the  very  close  coincidence  for  the  subjects  III  and  VI.  It  is 
of  importance  to  compare  the  results  of  the  observations  on  the 
just  imperceptible  negative  difference  with  those  of  the  obser- 
vations of  the  just  perceptible  negative  difference.  This  serves 
also  the  purpose  of  getting  material  for  answering  the  question 
whether  it  is  an  advantage  to  take  the  arithmetical  mean  of 
the  just  perceptible  and  of  the  just  imperceptible  difference. 


iubiect . 

Just 

percept] 

ible 

Just 

impercepti 

ible 

1    llTT^T'£M 

negati 

ve  differ 

ence. 

negative  difference. 

xyiiicrci 

I 

93.16 

93.82 

0.66 

II 

95.24 

94.71 

0.53 

III 

97.45 

98.32 

0.87 

IV  . 

95.05 

96.07 

1.02 

V 

94.22 

94.91 

0.69 

VI 

95.36 

95.04 

0.32 

VII 

96.04 

97.44 

1.40 

The  difference  between  these  quantities  are  small  and  in  this 
case  it  will  be  a  decided  advantage  to  take  the  arithmetical  mean 
as  the  representative  value,  as  it  is  prescribed  by  the  method  of 
just  perceptible  differences.  We  shall  make  the  following  con- 
siderations on  the  method  of  determining  the  threshold  as  the 
arithmetical  mean  of  the  just  perceptible  and  of  the  just  imper- 
ceptible differences.     Ther3  are  only  two  cases  possible:  either 


90  PROBLEMS  OF  PSYCHOPHYSICS 

the  stimulus    for    which    there    exists    the    probability    h    that 
a  "greater"  or  that  a  "smaller "-judgment   will  be  given  lies  in 
the  interval   between   the  empirical   determinations  of  the  just 
perceptible  and  of  the  just  imperceptil.)le  difference,  or  the  empir- 
ical determinations  lie  both  on  the  same  side  of  this  value.     The 
arithmetical  mean  comes  closer  to  the  true  value  than   at  least 
one  of  the  determinations  in  both  cases.     Add  to  this  that  it  is 
very  desirable  to  have  the  final  results  in  as  simple  a  form  as  pos- 
sible, and  there  will  remain  little  doubt  that  the  combination  of 
the  just  perceptible  and  of  the  just  imperceptible  difference  is  a 
very  serviceable  means  for  the  determination  of  the  threshold  in 
the  direction  of  increase  and  in  the  direction  of  decrease.     These 
facts  are  independent  of  the  theoretical  consideration  that  the 
just  perceptible  and  the  just  imperceptible  differences  are  empir- 
ical determination  of  the  same  quantity,  so  that  their  average  has 
the  character  of  being  the  most  probable  result.     The  two  fol- 
lowing tables  serve  the  purpose  of  showing  the  superior  accuracy 
of  the  mean  of  the  just  perceptible  and  of  the  just  imperceptible 
differences.     Table   78    (Appendix    p.  212  sq.)    is   a   composite 
table  of  the  individual  results  of  all  the  series.     The  results  for 
every  subject  are  given  in  one  group.     The  results  of  the  observa- 
tions on  the  just  perceptible  positive  difference,  on  the  just  imper- 
ceptible positive  difference,  on  the  just  perceptible  negative  dif- 
ference and  on  the  just  imperceptible  negative  difference  are  given 
on  one  line  for  every  series  of  100  experiments.     The  theoretical 
values  of  all  these  quantities  are  given  for  every  subject  in  the 
top  line  of  the  group.     The  column  marked  "Observed  values" 
contains  the  results  of  the  observations  and  the  numbers  opposite 
to  these  give  the  differences  between  the  observed  and  the  theoret- 
ical values.     These  differences  are  given  the  positive  sign  if  the 
observed  value  is  greater  than  the  theoretical  value,   and  the 
negative  sign  if  it  is  smaller.     The  averages  at  the  bottom  of  the 
columns  refer  to  the  differences  taken  regardless  of  sign.     The 
data  of  Table  79  (Appendix  p.  214)  are  arranged  in  a  similar  way, 
the  only  difference  being  that  the  numbers  do  not  refer  to  just 
perceptible  and  just  imperceptible  differences,  but  to  the  averages 
of  the  just  perceptible  and  just  imperceptible  difference,  i.  e.  to 
the  thresholds  in  the  direction  of  increase  and  to  those  in  the  di- 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  91 

rection  of  decrease.  The  comparison  of  the  averages  of  the  differ- 
ences between  the  theoretical  and  the  observed  values,  the  differ- 
ences being  taken  regardless  of  sign,  with  those  of  the  preceding 
table  shows,  indeed,  that  at  least  one  of  the  just  perceptil^le  and 
of  the  just  imperceptil)le  tlifferences  has  a  larger  sum  of  the  devia- 
tions of  the  observed  from  the  calculated  value  than  the  corres- 
ponding threshold.  The  threshold  in  the  direction  of  decrease, 
for  instance,  is  the  average  of  the  |ust  perceptible  and  of  the  just 
imperceptible  negative  difference.  For  subject  III  the  devia- 
tions of  the  observed  values  from  the  theoretical  value  of  the 
threshold  in  the  direction  of  decrease  are  on  the  average  0.495 
(Table  79),  which  is  smaller  than  0.5S0,  the  average  of  these  devia- 
tions for  the  just  imperceptible  negative  difference,  but  which  is 
greater  than  0.4SS  (Table  78),  the  average  of  the  deviations  for 
the  just  perceptiljle  negative  difference  for  the  same  subject.  It 
happens  in  four  cases  (the  positive  thresholds  for  the  subjects 
I,  II,  III  and  \ll)  that  the  sum  of  the  deviations  for  the  thres- 
hold is  smaller  than  any  one  of  those  for  the  corresponding  just 
perceptible  and  just  imperceptible  differences.  It  is  easy  to  see 
that  the  probability  of  this  event  is  equal  to  the  prol^ability  that 
the  theoretical  value  lies  in  the  interval  between  the  just  percep- 
tible and  the  just  imperceptible  difference,  and  that  it  is  nearer 
to  the  arithmetical  mean  of  the  observations  than  to  either  one 
of  them.  The  prol:)aliility  of  either  one  of  these  events  is  ^  and 
the  probability  of  the  compound  events  is,  therefore,  {.  Our 
results  show  that  this  event  happened  in  4  out  of  14  cases,  which 
is  exactly  what  we  have  to  expect.  This  coincidence  is  one  more 
argument  that  the  conditions  of  randomness  of  events  are  very 
closely  realized  in  our  experiments. 

The  final  evaluation  of  the  results  of  the  method  of  just  percep- 
tible differences  is  given  in  the  Tables  SO  and  81  (Appendix  p. 
215).  The  just  perceptible  and  the  just  imperceptible  differences 
are  combined  to  find  the  threshold  in  the  direction  of  increase 
and  decrease.  The  results  are  given  up  to  the  second  decimal 
place.  Table  80  gives  the  theoretical  results  and  the  following 
table  gives  the  observed  results.  The  coincidence  of  the  the- 
oretical with  the  observed  results  is  very  close  for  all  the  sub- 
jects except  for  subject  VII,  for  whom  it  is  less  satisfactory.     The 


92 


PROBLEMS  OF  PSYCHOPHYSICS 


two  thresholds  define  an  interval  outside  of  which  there  exists 
a  probability  exceeding  ^either  for  a  "heavier"  or  for  a  "lighter"- 
judgment.  We  may  express  this  fact  shortly  by  saying  that 
for  differences  beyond  the  threshold  there  exists  a  probability 
exceeding  ^  that  they  will  be  recognized.  Inside  of  this  interval 
one  can  not  expect  either  a  "heavier"  or  a  "lighter"-judgment 
with  an  even  or  more  than  even  probability,  and  for  this  reason 
it  is,  perhaps,  appropriate  to  call  this  interval  the  interval  of 
uncertainty.  We  give  here  a  comparison  of  the  theoretical  and 
of  the  observed  length   of  this   interval. 


Theoretical  value       Observed  value 


Subject, 

I 

II 

III 

IV 

V 

VI 

VII 


of  interval  of 

of  interval  of 

Difference 

uncertainty. 

uncertainty. 

6.15 

6.11 

-0.04 

3.96 

3.74 

-0.22 

1.43 

1.70 

+  0.27 

2.69 

2.6S 

-0.01 

2.67 

2.78 

+  0.09 

4.56 

5.12 

+  0.56 

4.07 

2.88 

-1.19. 

The  observed  values  exceed  the  theoretical  values  in  three  cases, 
and  they  fall  short  of  them  in  four  cases.  The  average  of  all  the 
deviations  (taken  regardless  of  sign)  is  0.34.  It  is  important  to 
have  an  idea  of  the  accuracy  of  the  determination  of  this  quan- 
tity, because  the  interval  of  uncertainty  is  the  basis  of  the 
psychophysical  measurements  afforded  by  the  method  of  just 
perceptible  differences,  as  it  is  used  to-day.  Indeed,  let  us  call 
the  intensity  of  the  stimulus  which  corresponds  to  subjective 
equality  r,  the  just  perceptible  positive  difference  r'^  and  the 
just  imperceptible  positive  difference  r''^.    We  then  have 


^o=i  (^o'  +  O 


and 


A^o  =  ''o-''- 


METHOD  OF  JUST  PERCEPTIBLE  DIFFERENCES  93 

Callius:  the  just  perceptible  negative  difference  r'^  and  the  just 
imperceptible  difference  r''^  we  find  in  the  same  way 

and 

The  quantities  A^u^nd  A^o  are  combined  as  a  rule  into  an  aver- 
age, which  is  called  the  mean  threshold  of  difference  and  which 
is  defined  by 

A'-  =  i(A'u+A''o). 
Introducing  the  values  r,  r^^,  r^  we  obtain 

A ''  =  H''  - '  u  + '  o- »')  =  H  ''o-  ^u) 

and  we  see  from  this  formula  that  the  mean  threshold  of  difference 
is  nothing  else  but  half  the  length  of  the  interval  of  uncertainty. 
The  method  of  just  perceptible  differences  prescribes  to  take 
this  quantity  as  a  measurement  of  the  accuracy  of  sensation, 
putting  this  accuracy  inversely  proportional  to  the  threshold 
of  difference. 

It  is  of  interest  to  notice  that  the  value  r  has  dropped  out  from 
the  formula  for  the  final  result  of  the  method  of  just  perceptible 
differences.  It  is,  therefore,  not  necessary  to  determine  it  and, 
as  a  matter  of  fact,  special  care  must  be  devoted  to  the  determi- 
nation of  the  point  of  subjective  equality  in  those  investigations 
where  only  the  threshold  in  the  direction  of  decrease  or  the  one 
in  the  direction  of  increase  is  determined,  or  where  the  amount 
of  overestimation  or  underestimation  is  of  interest.  Observa- 
tions on  the  point  of  subjective  equality  are  only  an  incident  in 
the  method  of  just  perceptible  differences.  There  is  nothing 
in  the  data  of  this  method  which  may  serve  as  a  definition  of  the 
objective  difference  of  stimuli  which  corresponds  to  subjective 
equality,  and  it  is  quite  impossible  that  this  method  gives  any 
indications  about  it,  because  its  algorithm  entirely  disregards 
the  "equality"-judgments  and  a  judgment  on  the  difference  which 
corresponds  to  subjective  equality  can  hardly  be  based  on  any- 
thing else  but  on  the  "equality"-judgments. 


94  PROBLEMS  OF  PSYCHOPHYSICS 

Our  discussions  show  that  the  result  of  the  experimental  pro- 
cedure which  is  called  the  method  of  just  perceptible  differences 
is  expressible  in  terms  of  probalnlities  of  judgments  belonging 
to  certain  types,  and  that  the  method  even  gains  in  lucidity  by 
doing  so.     These  judgments  are,  of  course,  either  correct  or  in- 
correct and  there  does  not  exist  a  fundamental  difference  be- 
tween this  method  and  the  error  methods  in  so  far  as  both  use 
the  same  data.     There  may  be,  however,  a  very  striking  difference 
in  so  far  as  the  attitude  of  the  subject  is  concerned.     The  results 
of  every  psychological  experiment  may  be  worked  out  in  two  ways. 
In  the  first  case  all  one  is  after  are  statistical  numbers  of  rela- 
tive frequency.     The  introspections  of  the  subject  are  of  little 
interest  and  they  are  of  importance  only  in  so  far  as  they  control 
the  correct  performance  of  the  experiments.     The-  only  state- 
ment ^yhich  the  subject  has  to  make  in  regard  to  his  mental  states 
is:  "I  complied  with  the  conditions  of  the  experiment."  The  sub- 
ject is  regarded  as  an  instrument  which  automatically  shows  when 
its  function  is  not  such  as  it  ought  to  be.     The  experimental  re- 
sults which  one  obtains  serve  for  the  determination  of  those 
factors  which  are  of  influence  on  the  function  of  the  subject  and 
the  analysis  of  this  function  is  the  only  purpose  of  the  inves- 
tigation.    The   second  w^ay  of  performing  psychological  experi- 
ments is  to  lay  stress  primarily  on  the  introspections  of  the  sub- 
ject.    The  numbers  of  relative  frequency  which   are  gained  in 
such  experiments  are  of  interest  merely  in  so  far  as  they  give  an 
indication  as  to  the  Constance  or    variability    of    the    conditions 
under  •  which  a  certain  psychical  state  arises.     It  may  seem  advis- 
able to  characterize  the  first  way  of  experimenting  as  the  psy- 
chophysical attitude,  and  the  second  as  the  psychological  atti- 
tude in  the  restricted  sense  of   the   w^ord.     Both   are   of  equal 
importance  and  neither  one  must  be  neglected  at  the  expense  of 
the  other,  but  confusion  is  inevitable  if  they  are  not  strictly  sep- 
arated.    If  one  uses  the  method  of  just  perceptible  differences 
as  a  purely  psychological  method  the  only  purpose  of  the  ex- 
perimental arrangement  is  to  help  the  subject  in  finding  a  stim- 
ulus on  which  he  may  give  the  judgment  that  it  is  the  smallest 
difference  which  he  can  perceive.     The   observations   which   are 
given   alongside  with  this  judgment  or  in  the  course  of  reaching 


.METHOD   OF   JUST    PERCEPTIBLE    DIFI'ERENCES  95 

this  judgment  are  by  far  more  important  than  the  numerical 
value  of  the  observed  difference.  The  variations  of  the  single 
observations  are,  perhaps,  more  interesting  than  the  final  result, 
because  they  give  an  indication  of  the  variability  of  the  men- 
tal state  under  observation.  The  purpose  of  the  experiments 
is  absolutely  different  if  one  uses  the  method  of  just  perceptible 
differences  as  a  purely  psychophysical  method.  In  this  case  one 
wants  to  form  one's  view  about  the  process  of  judging  differ- 
ences by  means  of  the  observed  numbers  of  relative  frequency. 
These  numbers  are  the  onl}^  outcome  of  the  experimentation  and 
they  must  be  put  to  the  best  possible  use,  since  it  must  be  re- 
quired that  all  is  got  out  of  the  results  what  is  in  them.  The 
error-methods  are  used  primarily  as  psychophysical  methods, 
whereas  the  method  of  just  perceptible  differences  may  be  easily 
adapted  for  both  puiposes. 

The  notion  of  a  just  perceptible  difference  was  the  object  of 
frequent  attacks  which  were  met  by  an  equally  stubborn  defence, 
because  this  notion  seemed  to  be  the  indispensable  basis  of  quan- 
titative psychology.  Fechner,  who  utilized  this  notion  for  this 
purpose,  had  given  it  the  signification  of  an  increment  of  sensa- 
tion thus  inaugurating  the  view  that  a  sensation  is  the  sum  of 
very  small  sensations.  This  view  in  connection  with  Weber's 
observation  of  the  Constance  of  the  relative  threshold  leads  to  the 
establishment  of  Fechner's  fundamental  formula  for  psychical 
measurement.  This  theory  combines  in  a  peculiar  way  the  feat- 
ures of  the  psychophysical  and  of  the  psychological  attitude. 
The  notion  of  a  difference  which  the  subject  just  perceives,  indeed, 
is  a  purely  psychological  one,  and  it  cannot  possibly  be  found  in 
the  statistical  numbers  of  relative  frequency,  unless  it  is  deprived 
of  its  psychological  character  and  transformed  into  the  notion 
of  a  difference  for  which  there  exists  the  probal^lity  h  that  it  will 
l)e  recognized.  There  is  no  objection  against  the  view  that  a 
sensation  is  constituted  of  elementary  sensations  from  the  psycho- 
physical point  of  view,  because  one  understands  in  this  case  b}^ 
sensation  nothing  but  a  certain  process,  which  may  very  well  be 
the  sum  of  other  processes.  The  question  merely  is  whether 
this  hypothesis  is  serviceable,  but  the  objection  that  a  sensation 
is  given  as  a  whole  and  not  as  the  sum  of  smaller  sensations  is 


96  PROBLEMS  OF  PSYCHOPHYSICS 

very  obvious  and  very  decisive  for  the  psychological  point  of 
view.  This  objection  was  raised  very  soon  against  Fechner's 
formula,  but  there  were  on  the  other  hand  certain  physiological 
facts  concerning  the  theory  of  sensations  which  spoke  in  favor 
of  this  view.  The  possibility  of  composing  sensations  of  their 
elements  seemed  to  give  the  possibility  of  exhausting  the  mani- 
foldness  of  sensations  by  a  limited  number  of  simple  sensations. 
For  these  simple  sensations  one  might  postulate  different  phys- 
iological processes  thus  reducing  the  study  of  sensations  to  that 
of  a  limited  number  of  elementary  processes.  The  principle  of 
specific  sense  energ}^,  thus,  became  more  or  less  consciously  a 
factor  in  favor  of  Fechner's  view.  At  the  same  time  the  notion 
that  a  sensation  is  the  sum  of  elementary  sensations  opened 
up  the  way  for  the  application  of  the  theory  of  errors.  The 
sensation  depends  only  on  the  combination  of  elementary  sen- 
sations and  it  depends  on  chance  which  combination  is  aroused 
by  a  certain  stimulus.  The  limits  inside  of  which  the  result 
may  be  expected  with  a  given  probability  determine  the  accu- 
racy or  precision  of  a  sensation.*  On  this  basis  one  may  try  find 
the  probability  of  correct  judgments  for  given  differences  of  the 
intensities  of  two  stimuli.  The  way  in  which  the  solution  of 
this  problem  was  attempted  may  be  illustrated  by  the  following 
example.  There  are  two  rifles,  A  and  B,  in  a  fixed  position  in 
such  a  way  that  A  is  aimed  at  a  point  which  is  somewhat  higher 
than  the  point  at  which  B  is  aimed.  This  difference  is  known. 
These  rifles  are  fired  a  great  number  of  times  and  one  observes 
in  each  case  whether  the  bullet  of  the  rifle  A  struck  a  higher  point 
than  that  of  the  rifle  B.  It  is  required  to  form  a  judgment  on  the 
precision  of  the  rifles  on  the  basis  of  these  data.     The  solution  of 

*The  expression,  "precision  of  sensation"  is  taken  from  Robert  MuellER 
Uehcr  die  Gmndlagen  der  Riclitigkeit  der  Sinnesaussagc,  Journal  j.  Psychologie 
u.  Neurologie,  Vol.  Ill,  1904,  pp.  112-126,  but  essentially  this  notion  was  used 
already  by  Fullerto.nt  and  C.\TTELL  /.  c  p.  12,  "Owing  to  the  complex  phys- 
ical, physiological  and  psychological  antecedents  of  the  perception,  the  same 
stimulus  is  not  always  accompanied  by  the  same  sensation.  There  is  a  nor- 
mal error  of  observation  ...."  These  authors  introduce  in  the  further  por- 
gress  of  their  discussion  the  hypothesis  that  this  normal  error  of  observation 
is  the  sum  of  elementary  errors,  a  hypothesis  which  leads  of  course  to  the  prob- 
ability integral. 


MKTHOD    OF    JUST    PEKCKPTIULE    DIFFEREN'CES  97 

this  problem  requires  the<t>  (;')-funotion  and  one  ol)tains  formulae 
wliich  are  analogous  to  those  used  in  the  method  of  right  and 
wrong  cases. 

The  agreement  with  the  theory  of  specific  sense  energy  and  the 
possibility  of  obtaining  a  safe  starting  point  for  the  quantita- 
tive methods  in  psychology  were  strong  arguments  in  favor  of 
the  view  that  a  sensation  is  built  up  from  sensational  elements. 
Our  considerations  show  that  it  is  fiot  necessary  to  conceive  of 
the  result  of  the  method  of  just  perceptible  differences  as  deter- 
mining a  sensational  element,  and  they  leave  to  this  difference 
merely  the  dignity  of  possessing  a  distinguished  numerical  value 
of  the  prol^ability  with  which  a  certain  class  of  judgments  is  passed. 
Our  analysis  shows  that  it  is  not  necessary  to  base  the  theory  of 
psychophysical  measurement  on  this  physiological  theory,  a  blow 
which  at  the  present  time  is  more  serious  for  this  theory  than 
for  the  deductions  of  quantitative  psychology. 

We  see  that  very  diverse  interests  came  into  play  in  the  prob- 
lem of  Fechner's  fundamental  formula.  In  this  formula  psy- 
chophysical and  psychological  views  are  intermingled  in  a  very 
peculiar  way,  and  it  seems  that  this  combination  of  view,  points 
which  are  essentially  different,  is  the  source  of  the  difficulties 
which  arise  in  the  interpretation  of  this  formula.  It  is  indispen- 
sable to  separate  the  psychological  and  the  psychophysical  attitude 
strictly  and  to  work  out  both  consistently.  The  statistical  num- 
bers of  relative  frequency  of  the  different  judgments  are  immedi- 
ately given  and  accessible  to  everybody.  The  sensations  of  the 
subject  are  not  known  to  anybody  but  to  himself,  and  we  have  to 
disregard  them  entirely  for  the  purpose  of  an  objective  treatment 
of  the  results  in  the  same  way  as  we  disregard  the  inner  states  of  a 
body  with  which  we  deal  in  physics.  These  considerations  show 
clearly  that  we  do  not  measure  sensations  in  psychological  ex- 
periments, neither  do  we  determine  the  relations  of  sensations 
to  certain  differences  of  stimuli.  What  we  determine  are  the 
probabilities  of  certain  events  and  the  nature  of  their  dependence 
on  certain  physical  conditions,  just  in  the  same  way  as  we  do  not 
measure  heat  but  certain  linear  magnitudes.  This  does  not  in- 
terfere with  our  being  able  to  solve  all  the  problems  of  psychophys- 
ics  just  in  the  same  way  as  if  we  could  measure  sensations.     All 


1 


98  PROBLEMS  OF  PSYCHOPH YSICS 

the  prol)lems  which  are  treated  in  quantitative  psychology  are 
accessible  either  Ijy  one  of  the  error  methods  or  by  the  method 
of  just  perceptible  differences,  or  by  both.  We  have  seen  that 
the  result  of  the  method  of  just  perceptible  differences  is  ex- 
pressible in  terms  of  probabilities,  and  so  are  the  results  of  the 
error  methods.  It  follows  from  this  that  every  problem  which 
can  be  treated  in  psychology  at  the  present  time  is  expressible 
in  terms  of  probabilities  of  the  different  judgments. 


i 


CHAPTER  IV. 

THE    EQUALITY    CASES; 

In  our  experiments  the  cases  in  which  a  "guess"-) udgment 
was  given  are  equivalent  to  equality  cases.  The  instructions 
given  to  the  subject  call  for  a  "guess"-judgment  when  no  dif- 
ference between  the  two  stimuli  could  be  detected;  there  ought 
to  be  nothing  in  the  sensation  to  go  upon  in  the  formation  of  the 
judgment  that  either  one  of  the  weights  is  heavier.  It  is  of 
some  interest  to  study  the  number  of  times  the  second  weight 
was  guessed  to  be  lighter  or  heavier  when  the  subject  believed 
to  be  unable  to  detect  a  difference.  We  will  study  the  dis- 
tribution of  the  guesses  between  the  "hg"  and  "l2:"-judgmentsby 
means  of  the  ratio  of  thenumberof  "hg"-judgments  tothe  number 
of  "Ig "-judgments,  because  only  the  relative  frequency  is  of  inter- 
est, and  because  this  ratio  allows  us  to  express  the  results  by  one 
number,  thus  reducing  the  size  of  the 'table  by  one-half.  The 
necessary  data  are  given  in  Tables  3-9  (Appedix  pp.  174-177). 
These  ratios  are  given  in  Table  82  (Appendix  p.  215)  for  all 
the  sul^jects  except  subject  III  for  each  pair  of  comparison  weights 
separately.  Numbers  of  one  column  refer  to  the  same  subject 
and  numbers  in  one  line  refer  to  the  same  pair  of  comparison 
weights.  The  results  for  subjects  III  are  not  given,  because 
this  subject  did  not  succeed  in  conforming  himself  to  the  condi- 
tions of  the  experiments  in  so  far  as  the  use  of  the  guesses  was 
concerned.  This  subject  gave  very  few  "guess"-judgments, 
and  those  which  he  gave  were  almost  exclusively  "hg"-judgments. 
An  inspection  of  Table  82  shows  that  the  numbers  in  the  upper 
third  of  the  table  are,  generally  speaking,  smaller  than  1  and 
that  they  are  in  the  lower  part  of  the  table  greater  than  1  with- 
out a  single  exception.  This  means  that  for  negative  differences 
the  "lighter-guesses"  are  in  the  majority  and  that  for  positive 
differences    considerably    more     "heavier-guesses"     are    given. 

99 


100  PROBLEMS  OF  PSYCHOPHYSICS 

There  exists,  therefore,  more  thap.  an  even  probabihty  that  a 
"guess "-judgment  will  be  correct.  This  seems  to  indicate  that 
the  subject  in  giving  a  judgment  on  the  equality  of  two  stimuli  is 
liable  to  overlook  certain  data  of  the  sensation  which  would  l)e  suf- 
ficient to  decide  his  judgment.  It  remains  undecided  whether 
this  is  generally  true  of  all  "ecjuality  "-judgments,  or  whether  it  is 
due  to  the  terminology  used  in  our  experiments.  This  termi- 
nology, indeed,  favors  the  mistaken  use  of  a  judgment  of  the 
equality  of  the  stimuli  given  with  all  positiveness  for  a  judgment 
of  a  very  faintly  perceived  difference.  With  all  the  subjects, 
however,  there  is  strong  introspective  evidence  that  the  "guesses" 
were  indeed  judgments  on  perceived  equality,  so  that  one  may 
favor  the  view  that  similar  conditions  will  be  found  also  if  an- 
other terminology  is  used.  It  is  important  in  the  choice  of  the 
terminology  by  which  one  allows  the  suliject  to  express  his  judg- 
ments to  consider  that  the  subject  has  the  natural  tendency  not 
to  commit  himself.  This  tendency  must  not  be  encouraged  by 
allowing  him  the  choice  of  a  type  of  judgment  to  which  he  may 
resort  as  a  subterfuge  for  judgments  given  with  a  low  degree  of 
confidence. 

It  was  seen  in  the  discussion  of  the  method  of  just  perceptible 
differences  that  the  threshold  of  increase  and  the  threshold  of 
decrease  enclose  an  interval  inside  of  which  neither  "heavier"- 
judgments,  nor  "lighter''-judgments  have  a  probability  equal 
to  or  surpassing  ^.  This  method  gives  no  indication  about  that 
difference  which  corresponds  to  subjective  equality.  It  is  evi- 
dent that  the  determination  of  the  point  of  subjective  equality 
must  be  based  on  the  equality  cases.  The  peculiar  difficulties  of 
this  problem  become  apparent  by  the  data  of  Table  83  (Appendix 
p.  215).  This  table  gives  the  numljers  of  relative  frequency  of 
the  " g"-judgments  for  every  pair  of  comparison  weights.  For  the 
subject  I,  for  instance,  these  numbers  set  in  with  the  frequency 
0.0622  for  84,  increase  constantly  to  0.4644  for  the  comparison 
of  100  gr  with  100  gr  and  after  having  reached  this  maximum 
they  drop  down  rapidly  to  0.0911  for  104  and  0.0533  for  108. 
Which  difference  shall  we  take  as  the  representative  value  for 
the  case  of  subjective  equality?  The  problem  thus  arising  is 
frequently   met   with   in   quantitative  psychology   and  in   other 


EQUALITY  CASES  101 

biometrical  sciences.  The  values  most  frequently  cliosen  as 
representative  for  tlie  entire  set  of  results  are  the  median,  the 
mode  and  the  arithmetical  mean.  It  is  obvious  that  the  median 
can  not  be  used  in  our  case,  because  the  range  of  the  differences 
used  is  determined  not  by  objective  conditions  but  by  the  chance 
element  of  subjective  choice.  It  would  be,  perhaps,  different 
if  this  range  -were  so  wide  that  the  proximal  and  the  distal  terms 
of  the  table  of  distribution  had  a  frequency  equal  to  or  near  to 
zero.  The  determination  of  the  mode  meets  with  considerable 
algebraical  difficulties  and  its  position  does  not  throw  much  light 
on  this  problem  as  will  be  seen  in  the  following  chapter.  There 
remains,  therefore,  only  the  arithmetical  mean  to  be  considered 
here. 

Let  us  make,  for  the  piu'pose  of  this  discussion,  the  following 
considerations.  Suppose  we  make  a  determination  of  the 
standard  weight  with  a  very  crude  balance.  The  set  of  weights 
with  which  we  have  to  compare  the  standard  weight  of  unknown 
intensity  may  consist  of  the  weights  84,  88,  92,  96,  100,  104,  and 
108  gr.  We  make  with  this  instrument  as  many  comparisons 
of  every  weight  with  the  standard  as  we  make  experiments  with 
a  subject.  Our  balance  will  show  in  some  cases  that  the  standard 
weight  is  heavier  than  the  comparison  weight,  in  other  cases  it 
will  be  found  to  be  lighter  and,  since  our  instrument  is  very  in- 
accurate, we  will  not  be  able  to  detect  any  difference  at  all  in  a 
certain  ntmiber  of  cases.  Those  cases  where  no  difference  can 
be  detected  are  noted  as  determinations  of  the  weight  of  the  stand- 
ard, and  the  discrepancies  between  the  single  observations  will  have 
to  be  eliminated  by  the  method  of  least  squares.  Making  these 
determinations  with  different  instruments  and  known  intensities 
of  weight  we  shall  obtain  results  which  not  only  allow  us  to  make 
a  statement  about  the  comparative  accuracy  of  the  balances, 
but  which  also  enable  us  to  tell  whether  our  instruments  are  affected 
by  a  constant  error.  Disregarding  entirely  the  "heavier"  and 
"lighter "-judgments  we  suppose  that  our  instrument  has  shown 
the  standard  to  be  equal  to  every  one  of  the  weights  as  many 
times  as  otir  subject  could  not  detect  a  difference  between  the 
weights.  The  total  numlier  of  experiments  is  the  same  in  both 
cases. 


102  PROBLEr-S  OF  PSYCHOPHYSICS 

A  set  of  results  arranged  for  this  purpose  is  given  in  Table  84 
(Appendix  p.  216).  The  columns  Nj^  give  the  number  of  times 
each  stimulus  seemed  to  be  equal  to  the  standard,  i.e.  was  ob- 
tained as  a  determination  of  its  weight.  These  results  must  be 
regarded  as  determinations  of  different  pondus*  of  the  same 
weight  and  the  column  under  the  heading  Nj^rj^  gives  the  number 
with  which  each  observation  comes  down  for  the  determination 
of  the  arithmetical  mean.  The  sums  of  these  numbers  are  given 
at  the  bottom  of  the  columns  in  the  line  marked  -T.  Dividing 
the  sum  of  the  Nj^rj^  by  the  sum  of  the  Nj^  one  obtains  the  most 
probable  value  for  the  determination  of  the  weight  of  the  stand- 
ard; these  numbers  are  given  for  each  subject  in  the  line  marked 
"Average".  One  then  proceeds  to  find  in  the  usual  way  the 
deviations  of  the  observations  from  the  mean  and  the  sum  of 
their  weighted  squares  obtaining  the  final  result  in  the  form: 


I 

96.10  ± 

1.44 

II 

96.43  + 

1.68 

III 

97.78  ± 

1.36 

IV 

97.15  ± 

1.52 

V 

96.98  + 

1.38 

VI 

97.65  + 

1.42 

vu 

98.03  A- 

1.58 

We  would  not  hesitate  to  say  that  instruments  which  give  such 
results  are  not  only  very  inaccurate,  but  that  they  are  also  af- 
fected by  a  considerable  constant  error.  The  difference  of  the 
general  mean  from  100  gr  would  have  to  be  taken  as  the  constant 
error  which  in  all  our  cases  is  negative,  so  that  we  have  clearly 
to  deal  only  with  subjects  who  overestimate  the  second  weight. 
The  comparison  of  the  accuracy  can  not  be  based  directly  on  the 
numbers  given  above  for  the  probable  error  of  the  mean,  because 

*The  use  of  the  English  term  "weight"  would  necessitate  constant  circum- 
locutions and  involve  the  possibility  of  a  mistake,  because  this  term  would 
be  used  in  two  different  meanings.  It  seemed  appropriate  for  this  reason 
to  use  the  Latin  term  whenever  the  word  "weight"  had  to  be  used  in  the 
meaning  which  it  has  in  the  theory  of  least  squares. 


EQUALITY  CASES  103 

the  different  series  are  not  equally  extended,  so  that  the  deter- 
minations for  the  different  subjects  have  not  the  same  pondus. 
The  relative  pondera  may  be  found  by  the  following  considera- 
tion. It  was  necessary  to  make  3150  experiments  with  subjeet 
III  in  order  to  obtain  16S  determinations  of  the  weight  of  the 
standard,  whereas  with  subject  VI  only  2100  experiments  were 
needed  to  obtain  401  determinations.  The  pondus  which  we 
give  to  the  different  determination;  depends  on  the  amount  of 
work  required  for  obtaining  the  result^,  and  since  all  the  experi- 
ments were  made  ^^•ith  equal  care,  we  put  the  pondus  propor- 
tional to  the  number  of  experiments  performed.  If  the  pondus 
of  the  determinations  for  the  first  three  subjects  is  1,  that  for 
the  subjects  I^'-^TI  will  be  only  §.  It  is  indifferent  whether 
we  base  our  comparison  of  the  accuracy  of  the  determinations 
of  the  weight  of  the  standard  by  the  different  subjects  on  the 
coefficient  of  precision  or  on  any  other  quantity  which  is  derived 
from  it.  We  choose  the  probable  error  of  a  single  determination, 
this  being  directly  comprobable  for  all  the  subjects.  The  values 
of  the  probable  errors  of  a  single  determination  for  the  different 
subjects  are  given  here  along  with  the  length  of  the  interval  of 
uncertainty  determined  above  by  the  method  of  just  percep- 
tible differences. 


Subject. 

Probable  Error. 

Interv 

'al  of  Uncertainty 

I 

38.37 

6.15 

II 

35.79 

3.96 

III 

17.69 

1.70 

IV 

26.69 

2.69 

V 

24.07 

2.67 

VI 

34.82 

4.56 

VII 

38.05 

4.07. 

These  results  show  that  in  general  a  large  probable  error  and  a 
large  interval  of  uncertaint}'  are  found  in  the  same  subject.  This 
correlation  will  become  clearer  if  we  arrange  our  seven  subjects 
first  by  the  size  of  their  probable  errors  and  then  by  the  length 
of  their  intervals  of  uncertainty.  Beginning  with  the  subject 
who  has  the  largest  probable  error  and  with  the  one  who  has  the 


104  PROBLEMS  OF  PSYCHOPHYSIGS 

greatest   interval   of   uncertainty    we   obtain   the   following   two 
arrangements. 


Order  of  the  subjects 

Order  of  the  subjects 

by  probable 

error. 

by  interval  of  uncertainty 

I 

I 

VII 

VI 

II 

VII 

VI 

II 

IV 

IV 

V 

V 

III 

^        III. 

This  little  table  shows  that  these  two  orders  go  parallel  to  a  cer- 
tain extent.  The  subject  who  appears  first  in  one  order  has  also 
the  first  place  in  the  second  order,  and.  the  subject  who  is  last  in 
one  order  has  the  same  place  in  the  other  order.  There  are, 
furthermore,  two  subjects  (IV  and  V)  who  appear  at  the  same 
places  in  both  arrangements.  The  other  three  subjects  appear 
at  different  places  in  the  two  arrangements,  but  subjects  II  and 
VII  appear  in  the  same  succession,  so  that  in  so  far  as  the  suc- 
cession is  concerned  only  subject  VI  is  out  of  his  place.  Sub- 
ject VII  is  on  second  place  and  subject  II  is  on  third  place  in  the 
order  by  the  size  of  the  probable  error,  and  these  subjects  have 
third  and  fourth  place  in  the  order  by  the  length  of  the  inter- 
val of  uncertainty.  This  means  that  only  one  out  of  seven  sub- 
jects appears  out  of  the  order  of  succession  in  the  two  arrange- 
ments. We  may  express  this  result  by  saying  that  the  size  of 
the  probable  error  in  general  goes  parallel  to  the  length  of  the 
interval  of  uncertainty. 

The  calculation  of  the  probable  error  is  based  on  the  sum  of 
the  weighted  squares  of  the  deviations  from  the  arithmetical 
mean.  G.  Fr.  Lipps  based  a  calculation  of  the  threshold  on 
the  same  value,  and  he  found  that  the  square  of  the  quantity 
which  he  calls  the  threshold  must  be  smaller  than  three  times 
the  sum  of  the  squares  of  the  deviations.  Our  results  bear  out 
in  a  general  way  that  this  quantity  may  be  used  as  a  measure- 
ment  of  sensations,    because   the   order    of  the    subjects   when 


EQUALITY  CASES  105- 

they  are  arranged  by  the  size  of  this  quantity  will  be  similar  to 
the  one  if  the  subjects  are  arranged  by  the  length  of  the  inter- 
val of  uncertainty.  The  same  holds  good  for  any  other  measure- 
ment of  sensations  which  is  based  on  the  sum  of  the  squares  of 
the  deviations  multiplied  by  a  constant  factor. 


CHAPTER  V. 

The  psychometric  functions. 

Until  now  we  treated  the  probabilities  of  the  judgments  for 
various  amounts  of  difference  in  the  intensities  of  the  stimuli 
as  unrelated  quantities.  It  is  a  very  natural  supposition  that 
these  probabilities  depend  on  the  amount  of  difference  between 
the  standard  and  the  comparison  stimulus.  We  immediately 
give  to  this  supposition  the  form  in  which  it  is  useful  for  further 
treatment  by  assuming  that  these  probabilities  are  analytic 
functions  of  the  difference  between  the  stimuli.*  There  exist, 
therefore,  three  different  functions  one  of  which  expresses  the 
probabilities  of  "lighter "-judgments,  the  second  the  probabili- 
ties of  "heavier "-judgments  and  the  third  the  probabilities  of 
"equality "-judgments  as  depending  on  the  amount  of  difference 
between  the  stimuli.  This  is,  indeed,  nothing  but  a  formulation 
of  the  fact  that  the  probabilities  must  depend  in  some  way  on 
the  amount  of  difference  between  the  stimuli.  We  have  to  ex- 
pect for  large  negative  differences  large  percentages  of  "  lighter  "- 
judgments  and  small  percentages  of  "equality "-judgments  and 
of  "heavier "-judgments.  In  a  similar  way  we  expect  with  a 
high  probability  "heavier "-judgments  for  considerable  positive 
differences,  for  which  differences  the  probabilities  of  ."equality  "- 
judgments  and  of  "heavier "-judgments  are  small.  We  make 
no  definite  hypothesis  about  the  nature  of  this  dependence  but 
we  merely  try  to  see  how  much  one  can  say  about  the  results  with 
the  minimum  addition  of  theoretical  suppositions.  The  func- 
tions which  give  the  dependence  of  the  probabilities  of  the  judg- 
ments on  the  differences  of  the  stimuli  may  be  called  the  functions 

*This  supposition  has  not  the  character  of  a  specific  hypothesis,  but  that 
of  a  general  supposition  necessary  for  mathematical  treatment.  A  dis- 
cussion of  the  importance  of  this  far  reaching  assumption  is  given  in  the  last 
chapter. 

106 


THE    PSYCHOMETRIC  FUNCTIONS  107 

of  the  distribution  of  these  judgments,  or  in  view  of  the  import- 
ance which  they  have  for  the  problems  of  quantitative  psychology 
one  may  call  them  the  psychometric  functions.  This  term  is  an 
imitation  of  the  term  "biometric  function"  which  is  in  common 
use  for  functions  which  represent  the  mortality  for  the  different 
ages.  There  are  as  many  psychometric  functions  as  one  admits 
types  of  judgments.  In  our  experiments  we  have  to  distinguish 
the  psychometric  function  for  the  "  heavier",  for  the  "lighter"  and 
for  the  "guess "-judgments.  The  number  of  psychometric  func- 
tions would  be  increased  considerably  if  the  degree  of  subjec- 
tive confidence  were  taken  in  consideration. 

Our  problem,  then,  is  formulated  in  this  way.  The  values 
of  a  function  were  observed  for  a  series  of  differences  and  the 
results  must  serve  for  the  determination  of  the  function;  what 
is  the  course  of  this  function  and  what  are  its  values  for  other 
differences?  The  answer  is  relatively  simple  if  the  nature  of 
the  function  is  known.  Such  a  function,  indeed,  must  depend 
on  a  certain  number  of  parameters,  which  can  be  directly  de- 
termined if  the  number  of  observations  is  equal  to  their  number, 
and  the  most  probable  values  of  which  can  be  determined  by 
the  method  of  least  squares  if  the  number  of  observations  ex- 
ceeds the  number  of  the  constants  of  the  function.  If  the  num- 
ber of  observations  is  smaller  than  the  number  of  parameters, 
the  problem  is  undetermined.  If  the  function  is  not  known,  one 
finds  the  values  for  other  differences  by  interpolation.  The 
interpolation  may  be  effected  either  by  means  of  Newton's 
method  of  differences  or  by  Lagrange's  formula  of  interpola- 
tion. Both  methods  are  based  on  the  assumption  that  a  set  of 
n  observations  can  be  represented  by  an  algebraic  function  of 
degree  (n-1),  but  even  if  this  supposition  is  not  justified  (the 
function  being  of  higher  than  the  (n-1)'^^  degree),  one  may  take 
the  values  given  by  these  formulae  as  values  of  the  simplest 
function  by  which  one  can  account  for  the  n  different  values.* 

*The  term  "the  simplest  function"  is  due  to  Gauss,  Theoria  intcrpola- 
tionis  mcthodo  nuva  iractata,  Werke,  Vol.  3,  p.  275  "Quae  praecedunt  sup- 
positione  innituntur,  functionem  X  ultra  potestatem  x"^*^  non  egredi,  sive 
differentiam  m  ^^™  una  cum  superioribus  evanescere,  in  quo  casu  methodus 
interpolationis  rigorose  vera  est.     Si  vero  ilia  suppositio  locum  non  habet, 


108  PROBLEMS  OF  PSYCHOPHYSICS 

The  results  of  our  experiments  when  brought  into  the  form  in 
which  they  are  given  in  Table  85  (Appendix  p.  217)  represent  sets 
of  seven  observations  on  the  three  psychometric  functions  for  each 
one  of  our  seven  subjects.  The  observations  are  made  for  the 
standard  weight  of  100  gr  and  with  the  seven  comparison  weights 
of  84,  88,  92,  96,  100,  104,  and  108  gr.  We  call  the  results  of 
these  observations  a^,  a^,  a^,  a^,  a^,  a^,  and  a^.  The  table  gives 
these  values  for  all  the  subjects  under  the  headings  G,  H  and 
L,  so  that  we  have  the  data  for  the  determination  of  all  the  psy- 
chometric functions.  The  general  expression  for  Lagrange's 
formula  is 

(X-X,)  (X-X3)  -.(x-xj  (x-x,)  (X-X3)  ■■■■(x-xj 

(Xi-X2)(Xi-X3)....(Xi-xJ^l        (X2-X,)  (X2-X3)  ....(jX-xj"'       •■*• 

^      (X-Xj)    (x-x,)  ....(x-x^.i)      ^  ' 


(X^-Xj)  (x^-x^)  ....(x„-xi^.j) 

where  x^   x,, x^^  are   the   values  for  which   the  observations 

were  made  and  a^,  a.^, aj^  are  the  results  of  these   observations. 

Introducing  the  data  of  our  experiments  inthisformula  we  obtain 

^      ,     (x-88) (x-92) (x-96) (x-100) (x-104) (x-108) 

^   \^)  : — z — .- — 7^ — -z^ — :r: a,- 


4.  8.  12.  16.  20.  24 
(x-84)  (x-92)  (x-96)  (x-100)  (x-104)  (x-108) 
4.  4.  8.  12.  16.  20 


a,+ 


(x-84)  (x-88)  (x-96)  (x-100)  (x-104)  (x-108) 
"~  8.  4.  4.  8.  12.  16 

(x-84)  (x-88)  (x-92)  (x-100)  (x-104)  (x-108) 


■«4  + 
1^.   O.   '±.    '±.   O.    1^ 

(x-84) (x-88) (x-92) (x-96) (x-104) (x-108) 
"^  16.  12.  8.  4.  4.  8.  ■  ^'" 

rx-84)  (x-88)  rx-92^  rx-96)  Tx-lOO^  Tx-lOS) 


20.  16.  12.  8.  4.  4 

_^  (x-84)  (x-88)  (x-92)  (x-96)  (x-100)  (x-104)  ^ 
24720.  16.  12.  8.  4  ^" 

interpolatio  eo  tantummodo  tendit,  ut  loco  functionis  X  functio  simplicissim 
eruatur,  per  quam  valoribus  A,  B,  C,  D,.,.  satifiat." 


THE  PSYCHOMETRIC  FUNCTIONS  109 

This  equation  of  sixth  degree  assumes  the  value  a,  for  x  =  84, 
the  value  a,  for  x  =  S8,  ...and  the  value  a^  for  x=108.  Intro- 
ducing for  X  any  particular  value  one  obtains  the  corresponding 
value  of  the  psychometric  function.  It  may  be  allowed  to 
make  the  following  remarks  in  regard  to  the  theoretical  and  prac- 
tical application  of  this  formula.  It  is  at  once  obvious  that  this 
equation  can  not  possibly  represent  the  psychometric  function 
in  its  entire  course.  This  function,"  indeed,  is  by  its  nature  of 
representing  a  mathematical  probability  limited  to  values  be- 
tween zero  and  one,  the  limits  included.  Values  greater  than 
the  unit  and  smaller  than  zero  can  not  be  admitted.  An  al- 
gebraic function  has  the  limit  co  for  x=  x  ,  i.  e.  it  assumes  very 
great  values  for  considerable  values  of  its  variable  and  one, 
therefore,  must  not  undertake  to  extrapolate  by  this  formula. 
The  occurrence  of  values  greater  than  the  unit  and  of  negative  | 
values  inside  of  the  realm  of  the  observations  has  only  sympto- 
matic significance. 

The  practical  application  of  Lagrange's  formula  is  extremely 
simple  and  easy  although  a  little  laborious.     It  is  a  matter  of 
course  that  one  must  arrange  the  computation  systematically 
whenever  one  has  to  treat  an  experimental  material  which  is  some- 
what extended.     In  this  respect  it  is  of  importance  that  the  a's  are 
the  same  throughout  the  whole  computation  for  a  psychometric 
function  of  one  subject.     The  interpolation  for  different  values  of  x 
will  be  effected  with  the  same  values  of  the  a's  but  with  different 
values  of  the  coefficients.     The  values  of  these  coefficients,  on  the 
other  hand,  do  not  depend  on  the  a's  and  remain  the  same  for  dif- 
ferent subjects.     It  is,  therefore,  possible  to  arrange  the  computa- 
tion in  two  ways,  if  one  has  to  treat  the  results  for  different  sub- 
jects with  whom  the  same  differences  were  used.     The  first  way  is 
to  attack  the  results  for  all  the  subjects  at  once,  making  the 
interpolation  for  one  value  of  x  for  all  the  subjects  at  the  same 
time.     The  other  way  is  to  carry  out  all  the  interpolations  for 
one  subject  before  beginning   those   for   another.     Every  inter- 
polated value  of  our  tables  is  the  result  of  seven  multiplications, 
two  additions  and  one  substraction.     This  work  is  reduced  con- 
siderably by  the  logarithmic  computation  which  consists  almost 
of  nothing  else  but  copying  logarithms,  which  were  looked  up  once 


no  PROBLEMS  OF  PSYCHOPHYSICS 

for  all,  and  performing  simple  additions,  if  the  computation  is  ar- 
ranged appropriately.  The  only  inconvenience  of  this  work  is  that 
there  are  no  thoroughgoing  checks,  so  that  the  calculation  must 
simply  be  done  over  again.  The  results  of  this  computation  are 
shown  in  the  Tables  86-92  (Appendix  pp.  218-221).  The  values 
of  the  psychometric  functions  for  the  "equality"  and  " heavier "- 
judgments  were  obtained  by  direct  interpolation  and  the  values 
for  the  "lighter"-judgments  were  found  by  subtraction  from 
'  one.  The  tables  show  the  values  of  the  psychometric  functions 
for  the  "heavier",  for  the  "lighter"  and  for  the  "equality"-] udg- 
ments  for  the  values  between  84  and  108  gr  with  intervals  of 
1  gr,  so  that  three  values  are  interpolated  between  two  observa- 
tions. The  column  under  the  heading  G  gives  the  values  of  the 
psychometric  function  for  the  "equality"-judgments,-and  the  let- 
ters H  and  L  at  the  head  of  the  third  and  second  column  stand  in 
abbreviation  of  the  psychometric  function  for  the  "heavier"  and  of 
that  for  the  "lighter"-judgments.  The  general  trend  of  these 
numbers  conforms  to  our  expectation  about  the  course  of  the  psy- 
chometric functions.  The  numbers  in  the  column  G  set  in  with 
low  values  and  gradually  rise  up  to  a  certain  maximum,  and  after 
this  is  attained  they  decrease  again.  The  numbers  H  set  in  with 
very  low  values  and  increase  at  first  slowly  then  more  rapidly 
assuming  ver}^  high  values  for  positive  differences.  The  mere 
inspection  of  the  tables  shows  two  striking  features.  The  first 
is  the  occurrence  of  negative  values  and  of  values  greater  than 
the  unit  in  the  results  of  all  the  subjects  except  for  the  sub- 
jects IV  and  VI.  These  breaks  occur  at  the  extreme  ends  of 
the  tables.  The  second  remarkable  feature  is  the  smallness  of 
the  values  in  the  column  G.  These  numbers  never  come  any- 
where near  the  unit  and  only  for  subject  I  do  they  approach 
the  value  ^.  For  none  of  the  other  subjects  do  they  exceed  the 
value  0.4  and  for  tw^o  subjects  they  do  not  reach  the  value  0.2 
(subjects  III  and  V).  This  confirms  our  previous  observation 
that  one  can  not  speak  in  any  absolute  sense  of  the  word  of  a 
point  of  subjective  equality,  because  for  this  value  one  would 
have  to  require  an  at  least  even  probability  for  the  "equality"- 
judgments.  There  remain  still  two  possibilities  of  defining  the 
point  of  subjective  equality.     In  the  first  case  the  probability 


THE   PSYCHOMETRIC  FUNCTIONS  111 

of  an  equality  judgment  is  greater  than  the  probability  of  either 
a  "greater"  or  a  "smaller"-] udgment  although  it  remains  smaller 
than  h  and,  therefore,  smaller  than  the  sum  of  the  probabilities 
of  the  "greater"  and  "smaller"-judgments.  The  interval  of  sub- 
jective equality  is  represented  in  this  case  by  those  differences 
in  the  intensities  of  the  stimuli  for  which  the  outcome  of  a  com- 
parison is  more  likely  to  be  an  "equality '-'judgment  than  either 
a  "greater"  or  a  "smaller"-judgment,  but  for  which  the  equality 
judgments  are  less  frequent  than  the  two  classes  together. 
There  still  remains  the  difficulty  of  defining  the  point  of  subjec- 
tive equality,  because  one  has  the  choice  between  the  centre  of 
the  interval  of  subjective  equality  and  the  value  for  which  the 
probability  of  an  '^equality"-judgment  is  a  maximum.  An  in- 
terval of  subjective  equality  exists  only  if  there  is  an  interval 
inside  of  which  the  psychometric  function  of  the  "equality "-judg- 
ments rises  above  the  psychometric  functions  of  the  "greater" 
and  of  the  "smaller"-judgments.  If  there  is  no  such  interval 
one  must  define  the  point  of  subjective  equality  as  the  value  for 
which  the  psychometric  function  of  the  "equality"- judgments 
attains  its  maximum.  This  definition  does  not  imply  that  "equal- 
ity"-judgments  are  more  probable  than  "greater"  or  "smaller"- 
judgments,  but  it  merely  says  that  for  this  point  the  judgments 
of  equality  are  more  frequent  than  for  any  other  difference  of 
the  stimuli.  It  is  not  necessary  that  there  exists  an  interval  of 
subjective  equality  in  which  the  psychometric  function  of  the 
"equality"-judgments  rises  above  the  psychometric  functions 
of  the  "greater"  or  "smaller "-judgments.  This  definition  of  the 
point  of  subjective  equality  is  the  most  general,  because  it  does  not 
make  any  assumption  about  the  course  of  the  psychometric  func 
tions. 

The  further  analysis  of  the  results  of  our  experiments  is  con- 
siderably facilitated  by  the  graphical  representation  of  the  psy- 
chometric functions  as  given  in  the  Charts  I-VII.  On  the  lines 
XX'  of  the  charts  the  intensities  of  the  comparison  stimuli  are 
represented,  the  unit  of  the  linear  measurement  corresponding 
to  1  gr.  The  ordinates  represent  the  probabilities  of  the  different 
judgments,  the  unit  of  the  Y  axis  being  chosen  ten  times  as  large 
as  that  of  the  abscissa.     The  lines  NN'  are  drawn  in  the  unit  of 


112 


PROBLEMS    OF    PSYCHOPHYSICS 


Chart  I. 


84 


88 


A  gQ  B/oo 

Chart  II. 


/04 


m 


THE    PSYCHOMETRIC    P^UNCTIONS 


113 


M" 


92  96  A        idoB 

Chart  III. 


92  A  96 

Chart  IV. 


114 


PROBLEMS  OF  PSYCHOPHYSICS 


~9l  ^4  B'96  "JSo 

Chart  V. 


-N' 


THE    PSYCHOMETRIC  FUNCTIONS 


115 


Chart  VII. 


distance  from  XX'.  The  psychometric  function  for  the  "lighter"- 
judgments  are  represented  in  each  chart  by  the  curve  passing 
through  the  point  A';  the  psychometric  function  for  the  "heavier"- 
judgments  is  represented  by  the  curve  passing  through  B',  and 
the  curve  passing  through  the  points  E  and  E'  represents  the 
course  of  the  psychometric  function  for  the  "equality"-]  udg- 
ments.  The  point  A  marks  the  values  of  x  for  which  the  psy- 
chometric function  for  the  "lighter"-judgments  assumes  the 
value  \;  A'  is  the  corresponding  point  on  the  curve.  The  point 
B  gives  the  intensity  for  which  the  psychometric  function  for 
the  "heavier"  judgments  assumes  the  value  h;  B'  is  the  corre- 
sponding point  on  the  curve.  The  lines  AA'  and  BB'  have  the 
length  h.  The  general  course  of  the  curves  is  the  same  for  all  sub- 
jects. The  tracings  show  at  once  whether  the  functions  assume 
negative  values  or  values  greater  than  the  unit,  because  in  these 
cases  the  curves  cross  the  lines  XX' or  NN'.  We  see,  furthermore, 
that  the  course  of  the  curves  is  in  most  cases  rather  irregular,  in- 
crease alternating  with  decrease;  only  for  the  subjects  II,  IV  and 
VI  does  it  come  near  to  regularity.  There  is,  however,  for  all 
the  subjects  an  interval  inside  of  which  all  the  three  psychome- 
tric functions  behave  regularly  and  this  interval  lies  approxi- 
mately in  the  middle  of  the  charts.  We  conclude  from  this  fact 
that  in  our  experiments  the  conditions  are  not  quite  the  same 
for  large  and  for  small    differences.     This  conclusion  is  in    per- 


116  PROBLEMS  OF  PSYCHO  PHYSICS 

feet  agreement  with  introspective  evidence  and  with  the  obser- 
vation of  previous  investigators  that  the  influence  of  attention 
in  series  of  comparison  stimuli  which  contain  only  small  differences 
is  different  from  that  in  series  which  contain  large  differences 
intermingled  with  small  ones.  The  points  for  which  the  psycho- 
metric functions  for  the  "heavier"  and  those  for  the  "lighter"- 
judgments  assume  the  value  J  are  always  inside  the  interval  in 
which  these  functions  are  regular.  A  close  inspection  of  the 
curves  in  these  intervals  shows  that  the  psychometric  functions  for 
the  "heavier"  and  for  the  "lighter "-judgments  differ  only  very 
slightly  from  straight  lines.  Perry's  "test  of  the  stretched  thread" 
discloses  scarcely  any  difference  of  the  curves  from  straight  lines 
in  these  intervals. 

This  circumstance  suggests  the  following  very  handy  deter- 
mination of  the  value  for  which  the  psychometric  functions  for 
the  "heavier"  or  for  the  "lighter "-judgments  assume  the  value 
+,  values  of  which  we  know  that  they  have  the  signification  of 
being  the  most  probable  result  of  the  threshold  in  the  direction  of 
increase  or  of  decrease  as  determined  by  the  method  of  just  per- 
ceptible differences.  The  complete  solution  of  this  problem  would 
require  to  solve  the  equation  F(x)=J.  The  work  of  solving  this 
equation  by  one  of  the  methods  of  approximation  is  insignifi- 
cant when  compared  with  the  amount  of  work  spent  in  setting 
up  the  equation  by  the  formula  of  Lagrange,  because  the  actual 
setting  up  of  the  equation  requires  a  great  number  of  long  mul- 
tiplications in  which  it  is  difficult  to  avoid  mistakes.  It  is  a  very 
fortunate  circumstance  that  the  conditions  for  finding  the  ap- 
proximate value  for  which  p  =  ^  are  such,  that  we  can  avoid  this 
long  computation.  One  sees  at  once  from  the  original  data 
given  in  Table  85,  in  which  interval  the  value  in  question  must 
be  sought.  One  includes  the  value  for  which  p  =  Mn  narrower  and 
narrower  limits  by  successive  interpolations,  until  the  interval 
is  small  enough  to  interpolate  on  a  straight  line.  The  charts 
show  that  it  is  not  necessary  to  go  in  our  results  beyond  inter- 
vals of  1  gr.  In  order  to  show  the  handiness  of  this  method  we 
will  illustrate  the  course  of  this  computation  by  finding  the  just 
perceptible  positive  difference  for  subject  V  from  the  data  of 
Table  85  without  having  recurrence  to  those  of  Table  90,  which 


THE    PSYCHOMETRIC  FUNCTIONS  117 

contains  the  values  of  the  psychometric  functions  for  this  sub- 
ject. The  data  of  Table  85  show  that  the  difference  for  which 
there  exists  the  probability  ^  for  a  "heavier"-judgment  must 
lie  somewhere  between  92  and  96.  The  numbers  of  relative 
frequency  of  the  "heavier "-judgments  (0.1600  for  92  and  0.5133 
for  96)  suggest  that  this  value  will  be  found  in  the  upper  part  of  the 
interval  nearer  to  96  than  to  94.  We  interpolate  by  Lagrange's 
formula  the  value  for  95.  Introducing  x  =  95  into  the  formula  given 
above  on  p.  IDS  we  find  the  following  values  of  the  coefficients: 

7.  3.-1.-5.-9.-13  273 

is  the  coefficient  of  ai  =  0.0267; 


4.  ^ 

i.  12.  16. 

20 

.24 

11. 

3.-1.-5 

.-9. 

-13 

4.. 

4.  8.  12. 

16. 

20 

11. 

7.-1.-5 

.-9. 

-13 

8. 

4.  4.  8.  ] 

[2. 

16 

11. 

7.  3.-5.- 

-9.- 

-13 

12. 

8.  4.  4.  8.  12 

11, 

7.  3.-1. 

-9.- 

-13 

16. 

12.8.4. 

4. 

8 

11. 

7.3.-1.- 

-5.- 

-13 

20. 

16.  12.; 

8.4 

:.    4 

11. 

7.  3.-1 

.-5. 

-9 

65536 

1287 
32768 

15015 
65536 

15015 
16384 


"  a2  =  0.0233; 
"  a3  =  0.1600; 
"  a,=0.5133; 


9009    ^, 

65536  "        "  -^  =  0-6800; 


1001 
32768 


"  ae  =  0.8700; 


231 

=_ "  "         "  a=0  94'^'^ 

24.  20.  16.  12. 8.  4         65536  '     ^-^^66. 

It  is,  perhaps,  worth  while  noticing  that  all  the  denominators 
are  powers  of  2.  This  is  a  consequence  of  our  stimuli  being  chosen 
equidistant  in  intervals  of  4  gr.  This  apparently  trivial  circum- 
stance facilitates  the  computation  not  inconsiderably,  and  it  is 
generally  recommendable  to  use  intervals  which  are  powers  of 
2.  In  some  cases  it  may  be  more  convenient  to  make  use  of 
the    method  of  differences  for  effecting  the  interpolation  and  if 


118  PROBLEMS  OF  PSYCHOPHYSICS  ^ 

one  has  to  deal  with  equidistant  values  distributed  over  an  inter- 
val which  is  a  power  of  2  one  can  apply  the  very  handy  formula 
for  the  "interpolation  in  the  middle." 

We  begin  conveniently  by  forming  the  sum  of  the  negative 
terms,  which  are  in  this  case  the  second,  fifth  and  seventh  terra. 
The  logarithmic  computation  may  be  arranged  thus: 

log  0.0233    =0.36736-2 
log      1287    =3.10958 


Sum     =1.47694 
log    32768    =4.51545 

Difference     =0.96149-4=  log  0.00092 

log  0.6800    =0.83251-1 
log      9009    =3.95468 

Sum     =3.78719 
log    65536    =4.81648 

Difference     =0.97071 -2  =  log  0.09348 

log  0.9433    =0.97465-1 
log        231    =2.36361 

Sum     =2.33826 
log    65536    =4.81648 

Difference     =0.52178 -3  =  log  0.00332. 
These  are  the  three  negative  terms;  their  sum  is 

0.00092 
0.09348 
0.00332 

Si  =  0.09772 


THE    PSYCHOMETRIC  FUXCTIONS  119 

By  a  similar  computation  one  may  find  the  positive  terms  and 
form  their  sum.  Ft)r  this  purpose  one  has  to  form  the  products 
for  the   first,  third,  fourth  and  sixth   term. 

log  0.0267    =0.42651-2 
log        273    =2.43616 


Sum  =0.86267 

log    65536  =4.81648 

Difference  =0.04619-4  =  log  0.00011 

log  0.1600  =0.20412-1 

log    15015  =4.17653 

Sum  =3.38065 

log    65536  =4.81648 

Difference  =0.56417-2  =  log  0.03666 

log  0.5133  =0.71037- 1' 

log    15015  =4.17653 

Sum  =3.88690 

log    16384  =4.21442 

Difference  =0.67248- 1=  log  0.47041 

log  0.8700  =0.93952-1 

log      1001  =3.00043 

Sum  =2.93995 

log    32768  =4.515^5 

Difference  =0.42450-2  =  log  0.02658. 


120  PROBLEMS  OF  PSYCHOPHYSICS 


This  gives  for  the  sum  of  the  positive  terms 

0.00011 
0.03666 
0.47041 
0.02658 


S,=   0.53376 

The  final  result  of  the  interpolation  is  found  by  subtracting  the 
sum  of  the  negative  terms  from  the  sum  of  the  positive  terms, 
which  gives 

§2=0.53376 

Si=  0.09772 


0.43604. 

We  find,  therefore,  that  the  value  of  the  psj^chometric  function 
of  the  "heavier "-judgments  for  95  is  for  this  subject  0.43604  or 
rather  0.4360  since  the  last  figure  is  not  exact.  This  is  the  value 
given  in  Table  90  (Appendix  p.  220);  all  the  values  given  in  the 
tables  were  found  by  a  similar  computation.  We  see  that  the 
value  of  the  psychometric  function  for  95  is  smaller  than  ^  and 
for  96  it  is  greater  than  h  The  value  of  x  for  which  the  psy- 
chometric function  assumes  the  value  h  lies  therefore  in  this  in- 
terval. The  difference  between  the  values  of  the  function  for  95 
and  96  is  0.0773  and  the  difference  of  0.4360  from  0.5000  is  0.0640, 
and  since  0.0640:0.0773  =  0.83  we  have  for  the  value  in  ques- 
tion the  approximate  result  95.83.  The  same  computation  may 
be  used  for  finding  the  just  perceptible  negative  difference.  These 
results  are  given  for  all  the  subjects  in  the  Tables  93  and  94  (Ap- 
pendix p.  221).  The  first  two  columns  give  the  observed  re- 
sults of  the  threshold,  and  the  values  calculated  by  the  algorithm 
of  the  method  of  just  perceptible  differences.  The  last  column 
is  marked  "Value  found  by  interpolation"  and  contains  the  re- 
sults calculated  in  the  way  just  described.  The  results  of  the  ob- 
servations and  the  values  calculated  by  the  different  methods 
are  sensibly  equal  in  all  the  cases.  The  differences  between  the 
observed  results  and  the  values  found  by  interpolation  are,  gen- 


THE    PSYCHOMETRIC  FUNCTIONS  121 

erally  spciikin*:,  iireater  than  those  between  the  observed  results 
and  the  values  found  by  the  algorithm  of  the  method  of  just  per- 
ceptil)le  differences.  The  value  found  by  interpolation  coincides 
only  in  the  case  of  subject  IV  so  closely  with  the  observed  result 
that  the  difference  is  consideral)ly  smaller  than  that  of  the  re- 
sult calculated  by  the  algorithm  of  the  method  of  just  percep- 
tible differences,  although  the  difference  between  the  latter  re- 
sult and  the  observed  result  is  only  0.16.  This  fact  that  the  re- 
sult of  the  method  of  just  perceptible  differences  coincides  better 
with  the  o])servations  is  not  surprising^  because  in  our  experi- 
ments only  one  set  of  differences  was  used  so  that  the  chance 
influence  of  the  choice  of  the  set  of  comparison  stimuli  is  very 
great.  To  this  fact  it  is  due  that  the  method  of  just  perceptible 
differences  gives  not  the  exact  values  for  which  p  =  ^  or  u  =  ^.  The 
results  of  the  interpolation  allow  us  to  determine  the  interval 
of  uncertainty  in  a  new  way;  we  give  these  quantities  alongside 
with  those  found  by  the  algorithm  of  the  method  of  just  percep- 
tible differences. 


Interval  of  uncertainty 

Interval  of  uncertaintv 

ibject. 

by  method  of  just  perceptible 
differences. 

by 

interpolation. 

I 

6.15 

7.69 

II 

3.96 

4.33 

III 

1.43 

1.57 

IV 

2.69 

3.02 

V 

2.67 

2.08 

VI 

4.56 

5.22 

vu 

4.07 

5.44. 

The  interpolated  values  deserve  more  credit  than  the  results 
of  the  method  of  just  perceptible  differences,  if  the  latter  is  not 
based  on  the  results  of  experiments  with  different  sets  of  com- 
parison stimuli.  The  values  found  by  interpolation  are  greater 
than  the  results  of  the  method  of  just  perceptible  differences  for 
all  the  subjects  except  for  subject  V.  It  is  not  void  of  interest  to 
order  the  subjects  by  the  size  of  the  interval  of  uncertainty  and 
to  compare  this  order  with  the  one  which  is  obtained  if  the  sub- 


122  PROBLEMS  OF  PSYCHOPH YHICS 

jects  are  ordered  by  the  size  of  their  probable  errors  which  were 
calcuhited  in  the  hist  chapter. 

Subjects  arranged  by  Subjects  arranged  by 

probable  error.  interval  of  uncertainty  found  by 

interpolation. 

I  I 

VII  \ll 

II  VI 

VI  II 

IV  IV 

V  V 

III  III. 

The  order  of  all  the  subjects  is  the  same  in  both -series  except 
for  the  subjects  II  and  VI  who  change  places,  The  probable 
errors  for  those  two  subjects  are  only  slightly  different  and  one 
may  account  for  this  break  in  the  regularity  of  the  two  series 
by  chance  disturbances  of  the  empirical  determination.  If 
this  explanation  is  accepted  one  comes  to  the  conclusion  that 
the  probable  error  derived  from  the  "equality "-judgments  by 
the  method  described  in  the  preceding  chapter  gives  a  measure- 
ment of  the  sensations  which  goes  parallel  to  that  afforded  by 
the  threshold. 

We  turn  now  to  the  study  of  the  psychometric  function  for 
the  "equality"-judgments.  The  curves  representing  these  func- 
tions comply  with  our  expectation  in  so  far  as  their  general  trend 
is  to  rise  at  first  and  then  to  decrease  after  having  attained  a  maxi- 
mum. They  have  in  common  with  the  curves  representing  the 
two  other  psychometric  functions  that  their  course  is  irregular 
in  the  extreme  parts  of  the  charts.  The  fact  that  the  curves 
do  not  rise  to  any  great  height  is  only  an  expression  of  the  ob- 
servation which  was  made  previously  that  the  relative  frequency 
of  the  equality  cases  is  very  low  even  in  that  interval  where  it 
is  greatest.  It  is  interesting  to  make  the  following  remark.  The 
points  E  and  E'  in  the  charts  mark  the  points  of  intersection 
with  the  curves  for  the  psychometric  functions  of  the  "lighter" 
and  of  the  "heavier"-judgments,  the  letter  E  being  given  to  the 
point  of  intersection  on  the  left.    The  curve  for  the  "equality'-' 


THE   PSYCHOMETRIC  FUNCTIONS  123 

cases  rises  above  the  curves  for  the  '  heavier"  an  J  for  tlie  "lighler"- 
jiulgments  in  those  cases  wliere  the  pohit  of  intersection  with 
the  curve  for  the  "lighter"-] udgnients  Hes  on  the  left;  this  state- 
ment is,  of  course,  restricted  to  points  of  intersection  which  lie 
withhi  the  interval  of  regularity.  The  point  of  intersection  of 
the  psychometric  function  for  the  "equality"-] udgments  with 
the  curve  for  the  "lighter"-judgments  lies  on  the  left  of  the  point 
of  intersection  with  the  curve  for  thq  ' heavier"-] udgments  only 
in  the  cases  of  the  subjects  I,  VI,  and  VII.  There  exists  for 
these  subjects  an  interval  of  subjective  equality,  which  in  the 
charts  is  marked  by  the  letters  E  and  E',  within  which  there  ex- 
ists a  greater  probability  for  an  "equality"-judgment  than  for 
any  other  kind  of  judgment.  One  must  make  the  distinction  be- 
tAveen  an  interval  of  uncertainty  and  an  interval  of  subjective 
equality.  All  subjects  have  an  interval  of  uncertainty,  but  only 
those  subjects  have  an  interval  of  subjective  equality  for  whom 
there  exists  an  interval  inside  of  which  the  probability  of  an 
''equality"-judgment  exceeds  the  probabilities  for  "heavier"  or 
"lighter"-j  udgments. 

The  curves  for  the  "equality"-judgments  approach  more  or 
less  the  type  of  a  bell  shaped  curve  inside  an  interval  which  is 
rather  extended.  The  curve  for  the  "equality"-j udgments  of 
subject  IV  resembles  as  much  a  probability  curve  as  one  may  ex- 
pect from  an  empirical  determination;  there  occurs  only  one  break 
at  the  extreme  upper  end  of  the  curve.  The  curve  for  subject  II 
comes  almost  as  near  to  the  type  of  a  probability  curve.  With 
the  exception  of  subject  I  all  the  curves  approach  the  type  of  a 
probability  curve  within  a  considerable  interval  and  even  for 
this  subject  the  course  of  this  curve  comes  very  near  to  that  of  a 
probability  curve  within  the  interval  from  98  to  104.  Outside 
the  interval  of  regularity  one  obtains  as  a  rule  values  which  are 
too  high,  a  fact  which  may  be  explained  by  the  remark  made 
above,  that  according  to  the  observations  of  previous  investiga- 
tors the  psychophysical  conditions  for  the  judgments  on  the  com- 
parison of  two  weights  are  not  the  same  in  series  which  contain 
only  small  differences  as  they  are  in  series  which  contain  large  and 
small  differences.  Variations  in  the  adaptation  of  attention  are 
possibly  the  cause  of  this  fact. 


124  PROBLEMS  OF  PSYCHOPH YSICS 

It  is  of  some  importance  and  not  void  of  interest  to  find  the 
maximum  value  of  the  psychometric  function  of  the  "equality"- 
judgments.  The  situation  of  this  maximum  throws  some  light 
on  the  question  of  what  we  have  to  understand  by  the  point  of 
subjective  equality,  and  the  maximum  value  of  the  psychometric 
function  of  the  "equality"-judgments  can  be  put  to  a  very  in- 
teresting use.  The  exact  determinations  of  the  value  of  x  for 
which  the  psychometric  function  attains  its  maximum  requires 
that  the  function  be  actually  set  up  by  Lagrange's  equation. 
The  maximum  is  found  by  differentiating  this  equation  and  solv- 
ing the  resulting  equation  of  fifth  degree.  This  process  is  very 
laborious  and  it  seems  to  be  advisable' to  use  the  following  meth- 
od. One  finds  in  the  tables  of  the  psychometric  function  of 
the  equality  judgments  the  three  values  of  x  for  which  the 
function  is  greatest.  Let  us  call  the  value  whicli  precedes  the 
maximum  A,  the  maximum  value  B  and  the  third  value  C,  and 
let  the  corresponding  values  of  x  be  x,,  x^  and  Xg.  These  values 
are  found  approximately  in  the  middle  of  the  tables  and  they 
are  always  in  the  interval  of  regularity,  so  that  all  the  values 
preceding  A  are  smaller  than  A  and  all  the  values  following  B 
are  smaller  than  B.     We  lay  through  these  three  points  a  parabola 

/(x)=ax'  +  fox  +  c 
in  such  a  way  that    x^is  the  origin  of  the  coordinates.     This  gives 

/(-1)=A 

/(0)=B 

/(I  =C 
which  is  s  system  of  three  linear  equations  for  the  determination 
of  the  three  coefficients  a,  b  and  c.     The  solution  gives 

A  +  C 


a  = 


h  = 


2 

C-A 


-B 


c  =  B. 

The  maximum  of  the  function'f  (x)  is  found  by 

/'(x,„)=2ax^  +  6  =  0"' 

which  gives  for  x,„ 

b 

^"'^~  2^ 


THE    PSYCHOMETRIC  FUNCTION'S  125 

or  if  we  express  the  values  of  a  and  1)  In-  A,  B  and  C 

A-C 


"'     2(A+C-2B) 

The  maximum  of  the  psychometric  function  is  then  determined 
by  Xj  +  Xj^.  This  value  lies  on  the  side  of  x,  or  X3  according 
to  whether  x„^  has  the  negative  or  the  positive  sign.  (1.)  The 
sign  of  x^  depends  on  the  sign  of  the  difference  A-C  if  B  is  the 
greatest  value,  and  the  maximum  .value  obviously  is  situated 
nearer  to  the  value  for  which  the  function  assumes  the  greater 
value.  (2.)  If  A  is  the  greatest  value,  the  difference  A-C  is  posi- 
tive and  the  sign  of  x^ depends  on  the  difference  A  +  C-2  B.  This 
difTerence  is  necessarily  negative  for  a  concave  function  decreas- 
ing, throughout  the  interval,  i.  e.  for  a  function  which  decreases 
more  rapidly  than  a  straight  line,  as  it  is  the  case  of  the  psychome- 
tric function  of  the  "equality "-judgments  in  the  interval  of  regu- 
larity for  values  of  x  which  are  greater  than  the  value  for  which 
the  maximum  is  attained,  x^  is  negative  and  the  maximum 
of  the  psychometric  function  lies  on  the  side  of  Xj.  (3.)  If  C  is 
the  greatest  of  our  three  values  the  difference  A-C  has  the  minus 
sign  and  A  +  C-2 B  must  be  negative, because  the  psychometric 
function  of  the  "equality"-judgments  is  an  increasing  concave 
function  inside  the  interval  of  regularity  for  values  of  x  which 
are  smaller  than  the  value  for  which  the  maximum  is  attained. 
Xj^  is  positive  in  this  case  and  the  maximum  of  the  psychometric 
function  lies  on  the  side  of  Xg. 

The  course  of  this  simple  calculation  may  be  illustrated  by 
working  out  the  results  for  sul)ject  I.  We  find  in  Table  86  that 
the  psychometric  function  of  the  "equality"-judgments  for  this 
subject  has  the  greatest  values  for  97,  98  and  99.  We  therefore 
have  to  put 

A  =0.4664  which  is  the  value  of  the  function  for  x  =  97, 
5  =  0.4827       "        "  "        "  "  "    x  =  98, 

C  =  0.4S46       "        "  "        "  "  "   x  =  99. 

These    quantities  serve  for  the  determination  of  x^^.     We  find 

from  these  data 

A-C  =  00182 

2(A+C -2  B)  =0.0288 

x,„=+0.63. 
The  maximum  of  the  psychometric  function  is,  therefore,  situated 
at  98.63. 


120  PROBLEMS  OV  PSYCHOPHYSICS 

It  is  of  some  importance  to  get  an  idea  about  the  degree  of  ap- 
proximation of  this  calculation.  It  is,  of  course,  possible  to  de- 
termine by  purely  theoretical  considerations  the  error  which  is 
due  to  the  approximate  representation  of  the  course  of  a  func- 
tion of  sixth  degree  by  a  parabola,  but  it  is  perhaps  best  to  show 
by  an  example  what  this  error  may  bo.  This  example  will  show 
that  a  considerable  amount  of  work  is  required  in  order  to  ob- 
tain a  determination  not  quite  as  good  as  ours,  if  the  maximum 
is  found  by  differentiating  the  equation  set  up  by  Langrange's 
formula.  This  is  due  to  the  fact  that  the  coefficients  of  the  equa- 
tion in  our  example  were  not  determined  accurately  enough,  be- 
cause if  they  had  been,  the  determination  of  the  maximum  would 
be  absolutely  exact.  This  example  serves  also  the  ulterior  pur- 
pose of  showing  that  the  application  of  Lagrange's  formula  is  a 
little  cumbersome  for  some  purposes  although,  or  rather  because, 
this  formula  represents  the  smallest  possible  theoretical  addition. 
The  idea  suggests  itself  that  a  more  definite  assumption  might 
lead  to  simpler  results.  It  seems  convenient  to  postpone  the  dis- 
cussion of  this  question  for  a  future  publication,  although  we 
might  be  in  the  possession  of  the  solution  of  this  problem. 

We  will  determine  the  position  of  the  maximum  of  the  psy- 
chometric function  of  the  "equality  "-judgments  for  the  subject 
I  by  setting  up  this  equation  by  the  formula  of  Lagrange.  The 
expression  of  Lagrange's  formula  has  the  form 

(x-88)  (x-92)  (x-96)  (x-100)  (x-104)  (x-lOS) 

F(x) .0.0622- 

4.  8.  12.  16.  20.  24 


(x-84)  (x-92)  (x-96)  (x-100)  (x-104)  (x-108) 
"  471l'8.  12.  16.  20 


.0.1244  + 


_      (x-84)  (x-88)  (x-96)  (x-100)  (x-104)  (x-108) 
8.  4.  4.  8.  12.  16 


(x-84) (x-88) (x-92) (x-100) (x-104) (x-108)    ^  ,,,^  . 

.0.4422 -h 

12.  8.  4.  4.  8.   12 


THE    PSYCHOMETRIC  FUNCTIONS 


(X-S4  (X-S8  (x-92)  (x-96)  (x-104)  (x-108) 

+   -^^ .0.4644- 

16.  12.  8.  4.  4.  8 

(x-84)  (x-88)  (x-92)  (x-96)  (x-100)  (x-108)     ^  ^^^ 
20.  16.  12^  8.  4.  4. 

(x-84)  (x-88)  (x-92)  (x-96)  (x-100)  (x-104) 

+  -^ ^  ~ -^ .0.0533. 

24.  20.  16.  12.  8.  4 

It  is  of  advantage  for  the  actual  setting  ii  ^  of  the  equation  to 
write  the  formula  in  this  way 

<^(x)  .  0.0622     0(x)    0.1244        0(x)   0.3311 
x-8F'2'^3^5"x-88  T'^.3.5  "^  x-92     ~2'\3    ~ 
(f>)x)    0.422        (/)(x)     0.4644       c6(x)      0.911 


+ 


x-96    2'^3'       x-100     2'^3         x-104     4^5,3.5 
^(x)     0.0533 


x-108    2'^3^5 
where 

.0(x)  =  (x-84)  (x-S8)  (x-92)  (x-96)  (x-100)  (x-104)  (x-108) 
Carrying  out  all  the  multiplications  indicated  gives 

<?!)(x)  =  x7-672x^+  193312x5-30858240x4 +  2952081664x^  - 
-169250070528x^  +  5384524382208x-73329396940S00. 

Dividing  by  the  linear  factors  one  obtains  the  expressions  needed. 

-^^^—  =  x^  -  588x5  +  143920x4  -  18768960x5  +  1375489024x^  - 
x-84 

-  5370899251 2x + 87296901 1200 

jHx)^  =x'  -  584x5  +  141920x4  -  18369280x^  +  1335585024x  '- 
x-88 

-  51718588416X  + 833288601600 
0(x)      _^6_  580x5  +  139952x4  -  17982656x^  +  1297677312x'- 


x-92 


-  49S63757824X  + 797058662400 


128  PROBLEMS  OF  P.SYCHOPHVSICS 

0(X) 


x-96 

x-100 

x-104 


=  x'^- 576x^  +  138016x4-  17608704x^  + 1261646080x^- 

-  48132046848X  + 763847884800 
=  x''-  572x5  + 1361 12x'^  +  17247040 \x  +  1227377664x'  - 

-  46512304128X  +  733293969408 
=  x^- 568x5  + 134240x4  -  16897280x'  +  1194764544x^  - 

-  4494557952X  +  705090355200 

-^M_  =  x^-5645x  + 132400x4-  16559040x^  +  1 163705344x^  - 
x-108 

-  43569893376X  +  678975897600 

For  the  purpose  of  setting  up  F  (x)  one  has  to  multiply  the  ex- 
pressions - — —  by  a^  and  to  divide  them   by  the  products  mj, 

X-Xj, 

where 

7nl  =  2'^3^5  a^  =  0.0622  x,=   84 

m2  =  2'5.3.  5  02  =  0.1244  X2=8S 

^3  =  2^^3  03  =  0.3311  X3=   92 

m,  =  2'4  32  a,  =  0.4422  x,=   96 

^5  =  2^^3  03  =  0.4644  X5=100 

^6  =  2^53.  5  06  =  0.0911  X6=104 

m,^2'\3\5  a,  =  0.0533  x,  =  108. 

These  multiplications  and  divisions  were  carried  out  to  the  18th 
decimal  place,  because  the  independent  variable  occurs  in  the 
sixth  power  and  is  to  assume  the  values  of  an  interval  around  100, 
and  because  it  was  intended  to  determine  the  values  of  the  func- 
tion exact  up  to  the  fourth  decimal  place.  The  next  step  is  to  form 
the  algebraic  sum  of  the  coefficients  of  the  different  powers  of 
X  which  is  done  conveniently  by  adding  up  the  positive  and  the  neg- 
ative terms  separately  and  forming  the  differences  of  these  sums. 
This  computation  is  given  on  the  insert  opposite  page  129.     The 

lines  are  marked    >^^^ ^and  contain  the  coefficients  of  the  powers 

mk(x-Xk) 
of  X    which  stand    at  the  head    of    each    column.      The  terms 


THE    PSYCHOMETRIC  FUN'CTIONS  120 

^tJS^'-         are  positive  and  the  term      '2k9v-     _  are   neg- 


X 

-  1132.778365833333333333  +  18411.8220 

83973.644081250000000000  1342295.9550 

109864.878525000000000000  1732084.7544 

787.446871250000000000  12271.2590 


195758.747843333333333333   3105063.7904 
165770.332198750000000000   2632256.2980 


-  29988.415644583333333333  +  472807.4924 

-  29988.415644583333333333 

X 

-  13089.584145000000000000  +   210899.0520 
144341 .302600000000000000  2290673.2850 

8339.445453750000000000  +    130683.8610 


165770.332198750000000000      +2632256.2980 


98.571  is  a  possible  maximum  of  the  function.  This  result  is  by 
0.06  smaller  than  that  of  our  approximate  determination.  These 
two  determinations  of  the  maximum  may  be  compared  with  the 
actual  course  of  the  function  in  the  interval  from  98.55  to  98.67. 
By  effecting  the  interpolation  in  the  usual  way  we  find  the  fol- 
lowing results 


a,^(x)  :  (X  -  84)m,  .000000021091037326 
a,<f>{x)  :  (X-  92 jm,  .000001684061686196 
a,<l>(x)  :  (x-lOO)TOs  .000002362060546875 
a,^(x)  :  (x-108)m,  .000000018073187934 


x'  X* 

.0000 1 240 1 52994791 6  + .0030354220980 1 3889 

.000976755777994789  .23568780 1 106770834 

.00 I 35 10986328 1 2500  .32 1 504785 1 56250000 

.0000 1 0 1 9327799479 1  .002392890082465277 


.393836835937500000  +     29.010490347222222222 

30.283901985677083333  2185.368642187500000000 

40.738552734375000000  2899.140356250000000000 

.299274641927083333  21.031865381944444444 


-  1132.778365833333333333  +     18411.8220 

83973.644081250000000000  1342295.9550 

109864.878525000000000000  1732084.7544 

787.446871250000000000  12271.2590 


1,  .000004085286458331 

2,  .000003437296549477 


.002350449218749996 
.001980424804687499 


.562620898443500000 
.474690136718759000 


71.717586197916666666 
60.586975781249999999 


5 1 34 .55 1 354 1 66666666666 
4342.969052864583333333 


195758.747843333333333333 
165770.332198750000000000 


3105063.7904 
2632256.2980 


Equation: 
Derivative: 


.000000647989908854 
.000003887939453124 


.0003700244 1 4062497  +  .08793076 1 724750000 
.001850122070312485  +.351723046899000000 


11.130610416666666666  +  791.582301302083333333 
33 .39 1 S3 1 250000000000   1 583 . 1 64602604 1 66666667 


299SS.415644583333333333 
2998S  .4 1 5644583333333333 


+  472807.4924 


a,^:x)  :  (  x-  88.)?n,  .000000253092447916 
a,<f,{x)  :{x-  96)m,  .000002998860677082 
a,^(x)  ■  (  X-  104)m,  .000000185343424479 


.000147805989583333 
.001727343750000000 
.000105275065104166 


+.035918880208333333 
.413890755208333333 
.024880501302083333 


4.649126041666666666  +  338.026483125000000000  -  13089.584145000000000000  +  210899.0520 
52.806050000000000000  +3783.500817708333333333  144341.302600000000000000  2290673.2850 
3.131799739583333333  +  221.441752031250000000    8339.445453750000000000   +  130683.8610 


.000003437296549477  -  .001980424804687499  +.47.4690136718750000   60.586975781250000000  +4342.969052864583333333  -  165770.332198750000000000  +2632256.2980 


THE    PSVCHOMETKIC  FUNCTIONS  129 


are  positive  and  the  term  ^     -  -  are  neg- 


mi'k  +  ,(x-x,k  +  i)  m2k(x-X2k) 

ative  and  their  coefficients  must  be  added  up  separately. 
These  sums  are  called  2\  and  J",-  ^^^^  differences  between  these 
sums  give  the  coefficients  of  the  equation,  which  are  given  in  the 
line  marked  "equation".    The  equation,  thus,  has  the  form: 

F(x)=0.000000647989908S54x^-    0.000370024414062497x5  + 
+  0.087930761724750000X*  -  11.130610416666667x5  + 
+  791.582301302083333  x^  -  29988.41 5644583333x  + 
+  472807.4924. 

If  the  coefficients  were  correct,  this  equation  would  assume 
the  value  ai  =  0.0622  for  x,  =  84,  a2  =  0.1244  for  X2  =  88,...  .  The 
values  of  the  function,  however,  are  exact  only  up  to  the  third 
decimal  place;  for  x=100,  for  which  value  the  substitution  is 
effected  most  conveniently,  one  finds  that  the  function  assumes 
the  value  0.4650  instead  of  0.4644.  Differentiating  F(x)  and 
putting  the  derivative  equal  to  zero  determines  the  value  of  x 
for  which  the  function  may  have  a  maximum.  The  coefficients 
of  the  derivative  are  given  on  the  insert  under  the  coefficients  of  the 
equation  to  which  they  correspond.  The  coefficient  of  x^'  of  the 
derivative  is  written  under  the  coefficient  of  x^"*"'  of  the  equa- 
tion.    The  derivative  has  the  form 

F'(x)  =0.000003887939453124x5  -  0.001850122070312485x^ 
+  0.351723046S990000x->      -  33.39183125000000x^ 
+  15S3.164602604166667X   -  29988.4156445833333 

Putting  this  equation  of  fifth  degree  equal  to  zero  shows  that 
98.571  is  a  possible  maximum  of  the  function.  This  result  is  by 
0.06  smaller  than  that  of  our  approximate  determination.  These 
tw^o  determinations  of  the  maximum  may  be  compared  with  the 
actual  course  of  the  function  in  the  interval  from  98.55  to  98.67. 
By  effecting  the  interpolation  in  the  usual  way  we  find  the  fol- 
lowing results 


130  PROBLEMS  OF  PSYCHOPHYSICS 


Value  of  the  psychometric 

Value  of  X. 

fun 

ctionof  the"equality"- 

jud 

gments  for  subject  I. 

98.55 

0.48600 

98.57 

0.48602 

98.59 

0.48605 

98.61 

0.48605 

98.63 

0.48604 

98.65 

0.48603 

98.67 

0.48602. 

This  little  table  shows  that  neither  98.57  nor  98.63  is  the  maximum 
of  the  function.  The  maximum  lies  in  the  interval  between  98.59 
and  98.61  nearer  to  the  second  value  than  to  the  first;  98.63  is,- 
therefore,  a  more  exact  determination  of  the  maximum  than  98.57. 
Such  a  difference,  however,  is  insignificant  for  our  purpose,  be- 
cause the  maximum  value  of  the  psychometric  function  of  the 
"equality"-]  udgments  is  of  greater  interest  than  the  value  of  x  for 
which  the  maximum  is  attained.  The  result  of  the  approximate 
determination  of  the  maximum  certainly  falls  in  the  neighbor- 
hood of  the  real  maximum,  and  the  determination  of  the  greatest 
value  which  the  psychometric  function  attains  is  not  considerably 
affected  by  a  small  error,  because  a  function  varies  only  little  in 
the  neighborhood  of  its  maximum. 

The  results  of  the  approximate  determination  of  the  position 
of  the  maximum  of  the  psychometric  function  of  the  equality  cases 
for  our  seven  subjects  are  given  here  alongside  with  the  arithmeti- 
cal means  of  the  "equality"-judgments  which  were  found  pre- 
viously. 

ean  Difference 

4-2.51 
+  1.14 
+  2.73 
+  0.19 
-1.09 
+  1.58 
-1.23. 


)ject 

Maximum 

Arithmetics 

I 

98.61 

96.10 

II 

97.57 

96.43 

III 

100.51 

97.78 

IV 

97.34 

97.15 

V 

95.89 

96.98 

IV 

99.23 

97.65 

Vll 

96.80 

98.03 

THE    PSYCHOMETRIC  FUNCTIONS  131 

The  maximum  comes  nearest  to  the  arithmetical  mean  for  the 
results  of  subject  IV;  this  is  only  a  consequence  of  the  fact  that 
the  curve  for  this  subject  is  very  regular  and  approaches  the  sym- 
metrical type.  It  is  interesting  to  notice  that  the  interval  inside 
of  which  the  maximum  values  vary  for  the  seven  subjects  is  by 
far  greater  than  the  interval  within  which  the  arithmetical  means 
vary.  The  smallest  value  of  the  maximum  is  95.89  and  the 
largest  is  100.51,  which  gives  4.62  for  the  length  of  the  interval. 
The  arithmetical  means  vary  between  96.10  and  98.01,  or  in  an 
interval  of  1.91.  The  values  of  x  for  which  the  psychometric 
function  for  the  "equality "-judgment  attains  its  maximum  are 
those  values  which  have  the  highest  probability  of  being  judged 
equal  to  the  standard  stimulus.  This  value  has  to  be  taken  as  a 
determination  of  the  point  of  subjective  equality.  It  is  of  interest 
to  notice  that  the  point  of  subjective  equality  does  not  coincide 
with  the  arithmetical  mean.  The  value  for  which  the  function 
reaches  the  maximum  is  in  five  cases  greater  and  in  two  cases 
smaller  than  the  arithmetical  mean;  the  difference  between  these 
two  values  is  1.50  on  the  average.  The  intensities  of  the  compar- 
ison stimulus  for  which  the  maximum  values  are  reached  are  gen- 
erally smaller  than  the  standard  weights  of  100  gr;  only  subject 
III  is  an  exception  since  for  this  subject  the  maximum  is  attained 
for  100.51.  This  indicates  that  the  second  weight  is  overesti- 
mated as  a  rule.  The  amount  of  overestimation,  however,  is 
smaller  if  we  take  the  value  for  which  the  "equality  "-judgments 
have  the  highest  probability  as  representative  of  the  point  of 
subjective  equality,  than  it  is  if  we  take  the  arithmetical  mean. 
The  next  step  is  to  find  the  maximum  value  of  the  function. 
This  value  is  found  by  introducing  the  value  of  x  for  which  the 
function  attains  its  maximum  into  Lagrange's  formula.  The 
values  of  x  for  which  the  psychometric  function  of  the  "equality" 
judgments  reaches  its  maximum  are  different  for  the  different 
subjects.  For  this  reason  the  computation  must  be  arranged 
in  a  way  slightly  different  from  the  one  described  above  and  it 
is  not  possible  to  attack  the  results  for  all  the  subjects  at  once. 
The  computation  is  considerably  facilitated  by  the  circumstance 
that  the  observations  are  made  for  all  the  subjects  on  the  same 
comparison  stimuli.     It  is  a  consequence  of  this  fact  that  the  con- 


182  PROBLEMS  OF  PSYCHOPHYSICS 

stant  denominators  of  the  terms  in  Lagrange's  formula  are  the 
same  for  all  the  subjects.     Let  us  call  these  denomiators  mj,m2, 

m^,  so  that  M;  is  the  constant  denominator  of  the  coefficient 

of  a;.  Inworkingout  the  results  for  subject  I  we  begin  by  find- 
ing the  logarithms  of  these  quantities  and  obtain 

log  m,  =  6.46969 
log  /«,  =  5.69154 
log  m3  =  5.29360 
log  m,  =  5.16866 
log  m,  =  5.29360 
log  mg  =  5.69154 
log  7/17  =  6.46969. 

We  have  found  by  our  abbreviated  method  that  the  psychometric 
function  of  the  equality  judgments  for  the  subject  I  reaches  its 
maximum  for  x'  =  98.63.     We  have  to  form  the  products 

(x'-88)  (x'-92)  (x'-96)  (x'-lOO)  (x'-104)  (x'-108) 
(x'-84)  (x'-92)  (x'-96)  (x'-lOO)  ■x'-104)  (x'-108) 
(x'-84)  (x'-88)  (x'-96)  (x'-lOO)  (x'-104)  (x'-108) 
(x'-84)  (x'-88)  (x'-92)  (x'-lOO)  (x'-104)  (x'-108) 
(x'-84)  (x'-88)  (x'-92)  (x'-  96)  (x'-104)  (x'-108) 
(x'-84)  (x'-88)  (x'-92)  (x'-  96)  (x'-lOO)  (x'-108) 
(x'-84)  (x'-88)  (x'-92)  (x'-   96)  (x'-lOO)  (x'-  104) 

The  simplest  way  to  find  these  products  is  to  form  the  product 

P=  (x'  -  84)  (x'  -  88)  (x'  -  92)  (x'  -  96)  (x'  -  100)  (x'-  104)  (x'  -  108) 

and  to  subtract  the  single  factors.  Introducing  the  value  98.63 
for  x'  we  obtain: 

log  14.63  =  1.16524 

log  10  63  =  1 .02653 

log    6.63  =  0.82151 

log    2.63  =  0.41996 

log    1.37  =  0.13672 

log    5.37  =  0.72997 

log    9.63  =  0.97174 

logP        =5.27167 


TUK  PSYCHOMETRIC  FUNCTIONS 


13:^ 


The  logarithmic  computation  takes  the  foUowinii  form: 

log  ai  =0.79379-2  log  m  ,      =6.46969 


logP   =5.27167 

I,    =4.06546 
2",    =7.63498 


log  14.63  =    1.16524 


i",   =    7.63493 


Ji-J,    =0.43053-4  =log  0.00027 

log  32  =0.09482-1 
logP   =5.27167 


I,  =4.36649 
I^  =6.71807 


0.64842- 3  =log  0.00445 

).51996- 
logP  =5.27167 


log  as  =0.51996-1 


I,  =4.79163 
J,  =6.11511 


=  0.67652-2  =  log  0.04748 


loga^  =0  64562-1 
logP  =5.27167 

I,  =4.91729 

I^  =5.58862 


1,-1,  =0.32867-1  =log  0.21314 

log  as  =0.66689-1 
logP    =5.27167 


I,  =4.93856 
J,  =5.43032 


logm,       =   5.69154 
log  10.63  =    1  02653 

i",  =   6.71807 


logmg     =5.29360 
log  6.63  =0.82151 

I,  =6.11511 


logm,     =5.16866 
log  2.63  =0.41996 

I.  =5.58862 


logm^     =5.29360 
log  1.37  =0.13672 

I,  =5.43032 


X-J,  =0.50824-1=  log  0.32229 


134 


PROBLEMS  OF  PSYCHOPHYSICS 


logae  =0.95952-2 
losP   =5.27167 


log  me     =-5.69154 
log  5.37  =0.72997 


Ji  =4.23119 
2,  =6.42151 

1,-1.  =0.80968-3=  log  0.00645" 

log  a;  =0.72673-2 
loffP    =5.27167 


2i  =3.99840 
I.  =7.44143 


I,  =6.42151 


logm^     =6.46969 
log  9.68  =0.97174 

J,  =  7.44143 


Ir^2  =0.55697-4  =  log  0.00036 

The  terms  resulting  from  a,,  a3  and  ag  have  to  be  taken  with  the 
negative  sign;  the  sum  of  these  terms  is 

0.00027 
0.04748 
0.00645 


So  =  0.05420. 


The  sum  of  the  terms  a,,  a^,  aj,  and  a^,  which  have  the  positive 
sign,  is 

0.00445 

0.21314 

0.32229 

0.00036 


S,  =0.54024 
S,  =  0.05420 

S,-  8,  =0.48604. 


The  difference  81-83  =  0.4860  is  the  value  which  the  psychome- 
tric function  of  the  "equality"-judgments  for  subject  I  attains 
at  the  point  x  =  98.63. 


THE  PSYCHOMETKIC  FUXCTIOIVS 


18o 


We  sive  here  tlie  results  for  all  our  subjects  indieatin.ti  the  value 
for  which  the  maximum  is  reached  and  the  value  which  the  psy- 
cliometric  function  attains  at  this  point. 


Subject 

I 
II 

III 

I\' 

V 

YI 

VII 


Position  of 

Maximum  value  of  the 

the  maximum 

psy 

chometric  function 

98.63 

0.4860 

97.57 

0.2667 

100.51 

0.1508 

97.34 

0.2111 

95.89 

0.1969 

99.23 

0.3771 

96.80 

0.3797. 

This  table  shows  clearly  that  the  values  of  the  psychometric  func- 
tion of  the  "equality"-) udgments  do  not  exceed  A  and,  except  for 
subject  I,  they  even  do  not  come  anywhere  near  this  value.  Let 
us  arran.se  our  seven  subjects  by  the  maximum  value  which  their 
psychometric  function  for  the  "eciuality"-judgments  attains,  and 
let  us  compare  this  order  with  those  which  we  obtain  when  we 
arrange  the  subjects  (1)  by  the  length  of  the  interval  of  uncer- 
tainty determined  by  the  method  of  interpolation,  (2)  by  the 
length  of  the  interval  of  uncertainty  determined  by  the  method 
of  just  perceptible  chfferences,  and  (3)  by  the  proliable  error  as 
determined  in  the  preceding  chapter. 


Order  of  subjects 
by  maximum  of 

the  psychometric 
function  of   the 
"G"-judgments. 

I 

VII 

M 

II 

IV 

V 

III 


Order  of  subjects 
by  interval  of 

uncertainty   de- 
termined by 
interpolation. 

I 

VII 
VI 

II 

IV 
V 

III 


Order  of  subjects 
by  interval  of  un- 
certainty deter- 
mined by  method 
of  just  percepti- 
ble differences. 

I 

VI 

VII 

II 

IV 
V 

III 


Order  of  sub- 
jects by  prob- 
able error. 

I 
VII 

II 

VI 

IV 

V 

III. 


136  PROBLEMS  OF  PSYCHOPHYSICS 

This  table  shows  that  the  order  of  the  subject  is  the  same  in  the 
first  and  second  column;  every  subject  appears  at  the  same  place 
in  these  two  columns.     We  conclude  from  this  fact  that  the  max- 
imum value  of  the  psychometric  function  may  seA^e  as  a  measure 
of  the  sensitivity  of  a  subject  just  as  well  as  the  interval  of  un- 
certainty as  determined  by  interpolation.     The  order  of  the  suli- 
jects  by  the  length  of  the  interval  of  uncertainty  as  determined 
by  the  method  of  just  perceptible  differences  and  the  order  by 
the  pro])able  error  are  slightly  different  from  those  preceding. 
All  the  four  orders,  however,  become  absolutely  identical  if  sub- 
ject VI  is  omitted.     This  subject  is  the  only  one  who  is  out  of  his 
place.     It  is  a  matter  of  course  that  the  determinations  of  the  in- 
terval of  uncertainty  by  the  method  of  just  perceptible  differences 
and  by  interpolation  must  give  similar  results,  because  both  meth- 
ods determine  the  same  quantity  and  their  results  must  coincide 
within  the  limits  of  the  accuracy  of  an  empirical  determination. 
It  is  more  surprising  that  these  results  coincide  with  those  obtained 
by  the  determinations  of  the  probable  error  and  of  the  maximum 
value  of  the  psychometric  function  of  the  "equality"-judgments. 
These  two  quantities  depend  entirely  on   the    "equality "-judg- 
ments, whereas  the  threshold  in  the  cUrection  of  increase  depends 
on  the  probability  of  "greater"-judgments   and  the  threshold  in 
the  direction  of  decrease  depends  on  the  probability  of  "smaller"- 
judgments.     The   threshold,   therefore,  does  not  depend  directly 
on  the  frequency  of  the    "equality"-judgments.     The  prol)able 
error  and  the  maximum  of  the  psychometric  function  of  the"equal- 
ity"- judgments,  furthermore,   are   found   by   algorithms    which 
have  nothing  in  common.     We  conclude  from  the  fact,  that  the 
results  of  these  four  different  methods  coincide,  that  there  must 
exist  some  kind  of  a  relation  between  the  values  of  the  different 
psychometric  functions,  and  there  must  also  exist  a  relation  be- 
tween the  probable  error  and  the  maximum  value  of  the  psy- 
chometric function  of  the   "equality"-judgments.     From    these 
facts  one  can  draw  certain  conclusions  as  to  the  nature  of  the  three 
psychometric  functions.     The  fact  that  the  length  of  the  inter- 
val of  uncertainty,  the  probable  error  and  the  maximum  of  the 
psychometric  function  of  the  "equality"-judgments  are  quantities 
each  one  of  which  may  serve  as  a  measure  of  the  sensitivity  of  a 


THE   PSYCHOMETRIC  FUNCTIONS  137 

subject  is  an  important  point  for  the  special  study  of  the  psycho- 
metric functions.     One  would  be  free  to  make  an  appropriate 
hypothesis  for  the  explanation  of  this  fact,  if  one  chooses  to  do 
so.     It  is,  however,  doubtful  whether  there  is  any  great  use  in 
introducing  a  hypothesis  at  this  point.     One  reason  for  introduc- 
ing a  hypothesis  consists  in  simplif\dng  the  practical  application 
of  a  method.     The  determination  of  the  limits  of  the  interval  of 
uncertainty  by  interpolation  and  the*  computation  of  the  probable 
error  are  so  handy,  that  it  is  not  likely  that  any  hypothesis  could 
simplify  them  very  much  or  define  a  measure  of  sensation  which 
is  more  readily  to  be  found.     Another  reason  for  introducing  a 
hypothesis  is  the  interest  which  we  take  in  connecting  different 
empirical  facts,  because  we  gain  a  new  insight  by  establishing 
relations  between  facts.     There  exists,  however,  no  such  need  at 
present  and  it  seems,  therefore,  to  be  best  to  leave  this  fact  for 
further  analysis  without  mixing  it  up  with  any  hypothesis.    Such  a 
hypothesis  would  refer  very  likely  to  the  nature  of  the  dependence 
of  the  probabilities  of  the  different  classes  of  judgments  on  the 
intensity  of  the  comparison  stimulus.     A  definite  assumption  on 
the  nature  of  the  psychometric  function  of  the  "equality"-cases 
is  indispensable  for  those  sciences  which  use  the  results  of  repeated 
observations  on  the  same  quantity.     Psychology  has  at  present 
no  practical  interest  in  making  such  an  assumption.     There  re- 
mains the  theoretical  interest  in  the  problem,  which  can  be  satis- 
fied only  by  the  results  of  further  analysis  and  not  by  a  hypothesis. 
The  special  study  of  the  psychometric  functions  will  be  the  ob- 
ject of  a  future  publication.     We  may  sum  up  the  peculiarly  ad- 
vantageous position  of  psychology  by  saying  that  the  dependence 
of  the  probabilities  of  the  different  classes  of  judgments  on  the 
intensity  of  the  comparison  stimulus  is  an  object  of  investigation 
for  psychology,  and  that  it  need  not  be  made  the  object  of  a  hy- 
pothesis.    This  problem  can  be  treated  only  accidentally  in  other 
sciences,  becau.se  all  that  is  needed  there  is  a  fixed  standard  by 
which  the  accuracy  of  different  determinations  can  be  compared, 
and  a  definite  algorithm  for  the  combination  of  o]:>servations  which 
do  not  give  identical  values  for  the  same  quantity. 

We  have  seen  that  the  probable  error  and  the  maximum  prob- 
ability for  an  "equality "-judgment  may  be  used  for  a  measure- 


138  PROBLEMS  OF  PS YCHOPHYSICS 

ment  of  the  accuracy  of  sensation  in  the  same  way  as  the  inter- 
val of  uncertainty.  These  quantities  are  entirely  based  on  the 
probabihties  of  the  "equality  "-cases.  This  fact  is,  perhaps,  a  lit- 
tle surprising  if  one  remembers  that  this  class  of  judgments  was 
more  or  less  of  a  difficulty  for  the  psychophysical  methods,  a 
difficulty  which  seemed  to  be  so  serious  that  some  investigators 
recommended    suppressing  these  judgments  entirely. 


CHAPTER  VI. 

A   GENERA  f.   INQUIRY   CONCERNING  THE    PSYCHOMETRIC   FUNCTIONS. 

We  have  avoided  introducing  into  the  alwve  discussion  any 
one  of  those  notions  which  in  most  of  the  current  text-books 
of  psvcholog}"  are  proclaimed  as  indispensable  presuppositions  of 
this  science.  There  is,  for  this  reason,  no  necessity  for  philosophi- 
cal considerations  in  our  argumentation.  The  notions  of  which 
we  have  made  use  are  by  no  means  different  from  those  used 
e.  g.  in  the  theory  of  life  insurance  or  in  the  formal  theory  of 
population.  There  is,  however,  one  point  in  our  considerations 
which  seems  to  throw  considerable  light  on  the  epistemological 
nature  of  .some  scientific  methods  and  which  for  this  reason  de- 
serves some  attention. 

The  point  in  question  is  the  introduction  of  the  assumption 
that  the  probabilities  of  the  different  judgments  are  analytic 
functions  of  the  amount  of  difference  of  the  intensities  of  the 
stimuli.  This  assumption  leads  to  the  construction  of  the  notion 
of  a  psychometric  function  which  was  found  useful  for  many  pur- 
poses. We  confined  our  discussions  to  the  considerations  of  the 
case  that  an  algebraic  function  of  sixth  degree  expresses  the  re- 
sults of  our  observations  on  seven  values,  because  this  assump- 
tion is  the  smallest  possible  theoretical  addition.  The  degree  of 
this  equation  depends  merely  on  the  number  of  the  values  for  which 
observations  were  made  and  the  mathematical  expression  may 
be  adapted  to  any  number  of  observations  whatsoever.  Inter- 
polating by  Lagrange's  formula  has  not  the  character  of  a  definite 
hypothesis  on  the  nature  of  the  psychometric  function,  but  it  is 
rather  a  means  of  completing  a  set  of  observations.  The  method 
of  expressing  a  set  of  n  observations  by  an  algebraic  function  of 
degree  n-1  lends  itself  more  readily  to  different  results  than  any 
other  hypothesis;  it  is  of  course  indifferent  whether  the  values 
of  these  functions  are  determined  by  Lagrange's  formula  of  inter- 

139 


140  PROBLEMS  OF  PSYCHOPHYSICS 

polation  or  whet lier  Newton's  method  of  differences  is  used.  We 
have  seen,  on  the  other  hand,  that  an  algebraic  function  can  be 
only  a  preliminary  result,  because  it  cannot  possibly  give  the 
dependence  of  the  probabilities  of  the  judgments  of  different 
types  for  all  the  values  of  the  comparison  stimulus,  since  an  al- 
gebraic function  must  assume  values  greater  than  the  unit  or 
smaller  than  zero.  The  question  as  to  the  nature  of  this  depend- 
ence of  the  probabilities  on  the  intensity  of  the  comparison  stim- 
ulus can  be  settled  definitively  only  by  experimental  evidence; 
as  long  as  no  facts  are  at  hand  one  is  free  to  introduce  anv  hy- 
pothesis which  may  seem  appropriate.  The  following  consider- 
ations will  show  that  some  caution  is  recommended,  because  the 
assumption  of  a  definite  form  of  the  psychometric  function  is  a 
far  reaching  hypothesis. 

The  immediate  result  of  the  observations  is  a  set  of  results  for 
a  greater  or  smaller  number  of  observed  differences.  The  num- 
ber of  observations  must  be  of  course  sufficient  to  determine  the 
constants  of  the  psychometric  function  the  form  of  which  is  sup- 
posed to  be  known;  if  the  number  of  observations  is  greater  than 
the  number  of  parameters  in  the  psychometric  functions  one  may 
combine  the  results  and  determine  the  most  prol^able  value  of 
the  constants,  but  the  problem  remains  indeterminate  if  the  num- 
ber of  observations  is  smaller  than  the  number  of  constants  of  the 
function.  After  the  constants  of  the  function  are  found  one  can 
give  the  probabilities  for  all  the  intensities  of  the  comparison 
stimulus.  This  means  that  a  finite  number  of  observations — 
in  many  cases  a  very  small  number — is  sufficient  to  serve  as  a 
basis  for  an  infinity  of  statements.  The  probabilities  of  the  dif- 
ferent classes  of  judgments  depend  on  the  psychophysical  condi- 
tion of  the  subject  under  observation,  and  in  the  absence  of  more 
complete  information  we  can  characteiize  the  psychophysical 
condition  of  a  subject  only  hy  the  probabilities  with  which  the 
different  classes  of  judgments  may  be  expected  for  various  differ- 
ences. It  would  seem  that  an  infinite  number  of  observations 
were  needed  for  this  purpose,  because  there  is  no  reason  a  priori 
why  the  group  of  causes  which  gives  a  certain  probability  to  the 
judgment  "greater"  on  the  difference  x  should  be  in  any  connec- 
tion with  the  group  of  causes  which  determine  this  probability 


THE    PSYCHOMETRIC    FUNCTIONS  141 

for  the  difference  y.  The  psychophysical  conditions  of  a  subject 
which  determine  the  probabilities  of  the  judgments  of  different 
classes  on  a  given  difference  come  obviously  under  the  heading 
of  what  is  usually  called  an  attribute  or  a  quality.  The  existence 
of  a  psychometric  'function  which  can  be  determined  by  a  finite 
number  of  observations  implies  that  there  exist  certain  relations 
between  the  different  qualities,  so  that  all  the  qualities  can  be 
expressed  by  a  certain  number  of  them.  It  is  obvious  that  no 
experience  whatsoever  can  be  the  basis  of  such  a  statement,  and 
it  follows  from  this  that  the  notion  of  a  psychometric  function 
is  not  a  result  of  experience,  but  the  expression  of  the  methodolog- 
ical assumption  that  there  exist  relations  of  some  .kind  between 
the  dift'erent  qualities  of  the  subject.  There  is  no  doubt  that  the 
conclusions  drawn  from  this  assumption  are  in  agreement  with 
experience  and  one  therefore  can  only  ask:  Why  is  it  that  one 
can  conclude  from  one  quality  of  an  object  to  another  quality? 
Why  must  an  object  have  certain  qualities  because  it  has  certain 
other  qualities?  The  formal  character  of  the  logical  process  is 
that  of  a  subsumption.  The  general  expression  of  the  psychometric 
function  is  the  major  which  is  specified  by  introducing  special 
values.  The  difficulty  is  not  of  logical  but  of  epistemological 
nature  aiid  refers  to  the  way  in  \vhich  the  psychometric  function 
is  established.  It  is  quite  obvious  that  we  never  can  reach  a 
proposition  of  such  generality  by  induction,  because  the  psycho- 
metric function  makes  a  statement  about  every  element  of  a  class 
which  is  of  the  order  of  the  continuum,  and,  as  a  matter  of  fact, 
the  number  of  observations  on  which  this  statement  is  based  is 
rather  small.  It  is,  therefore,  not  a  generality  of  the  empirical 
type.  It  is,  furthermore,  clear  that  the  relations  which  exist  be- 
tween the  different  qualities  of  a  subject  are  not  of  the  causal 
type,  if  this  word  is  used  in  its  common  meaning  which  refers  to  a 
succession  of  phenomena.  Qualities  of  an  object  are  simultane- 
ous, not  successive,  and  the  relation  of  qualities,  therefore,  can- 
not possibly  be  of  the  causal  tj^pe.  It  does  not  avail,  either,  to 
take  the  view  that  the  qualities  which  determine  the  probabilities 
of  a  judgment  of  a  certain  type  for  different  intensities  of  the 
comparison  stimulus  must  follow-  a  certain  law,  because  they  are 
qualities  of  the  same  sul)ject.     The  only  rigorous  condition  that 


142  PROBLEMS  OF  PSYCHOPHYSICS 

must  be  satisfied  by  qualities  of  the  same  object  is  that  they  are 
not  contradictory.  There  is  nothing  in  the  quahties  which  deter- 
mine the  prol^iliihty  of  a  "heavier "-judgment  for  the  difference  x 
which  impHes,  that  there  must  be  a  certain  other  probabihty  for 
another  difference  y.  Such  a  conchision  can  only  be  made  after 
the  dependence  between  these  qualities  is  established  l)y  means 
of  the  psychometric  function,  and  one  will  tr}^  in  vain  to  formu- 
late the  problem  in  such  a  way,  that  the  probability  of  a  "heav- 
ier "-judgment  for  the  difference  y  follows  from  that  for  the  differ- 
ence X  In-  logical  processes,  if  one  does  not  use  the  notion  of  a 
psychometric  function. 

This  problem,  which  is  decidedly  of  philosophic  nature,  is  little 
treated  in  the  literature  and  it  does  not  seem  that  anybody  has 
given  a  solution  besides  that  of  Kant.  This  solution  may  be 
formulated  in  this  way.  The  empirical  datum  are  certain  ob- 
served values  which,  in  themselves,  are  not  a  law  nor  are  they  in 
relation  to  each  other.  Such  a  relation  can  be  established  merely 
by  representing  these  data  in  a  possible  intuition.*  The  empiri- 
cal representative  of  this  intuition,  in  our  case,  is  supplied  by 
representing  the  results  by  points  in  a  plain  and  by  a  curve  drawn 
through  them.  The  representation  in  a  pure  intuition  makes 
it  possible  to  find  a  law  of  the  generality  of  the  psychometric 
functions.  Without  trying  to  refute  or  to  confirm  Kant's  solu- 
tion we  will  try  to  show  by  the  following  considerations  that  the 
question  of  the  relation  between  the  qualities  of  an  object  implies 
a  problem  of  verj^  great  generality,  which  contains  the  epistemolog- 
ical  problem  of  the  principle  of  causality  as  a  special  case. 

The  totality  of  those  facts  which  constitute  experience  are  the 
material  which  science  works  on.  There  is  no  primary  dis- 
tinction between  these  facts,  all  of  which  stand  on  the  same  level 
of  reality,  but  a  great  number  of  secondary  distinctions  comes 
in,  which  are  introduced  for  the  sake  of  the .  description  of 
the  phenomena,   because   the  task  of  giving  an   exhaustive  de- 

*The  word  intuition  has  become  customary  as  a  translation  of  the  German 
term  "  Anschauung, "  but  this  translation  is  not  quite  appropriate  in  so  far 
as  this  word  carries  the  side  meaning  of  "as  if  by  inspiration".  For  this 
reason  it  would  he  advisable  to  invent  a  new  word  for  the  translation  of  this 
purely  technical  terra. 


THE    PSYCHOMETRIC    FUNCTIONS  HH 

scription  is  made  easier  by  classifyiiifz;  the  events  into  (groups. 
The  term  "science"  is  frequently  used  in  two  slightly  different 
significations.  In  the  first  signification  this  term  designates  a 
group  of  propositions  which  describe  a  certain  part  of  the  realm 
of  experience,  but  in  its  second  meaning  this  term  is  applied  to 
every  effort  or  activity  which  has  the  purpose  of  establishing 
such  a  system.  We  shall  speak  of  science  only  in  the  first  signi- 
fication as  applied  to  a  set  of  propositions  which  describe  a  given 
part  of  experience.  We  may  speak  of  a  system  of  science,  if  its 
single  propositions  are  connected  in  such  a  way  that  all  the  prop- 
ositions may  be  deduced  from  a  set  of  propositions.  We  may 
speak  in  this  case  of  a  systematic  science  in  contra-distinction  to 
descriptive  sciences  which  may  be  able  to  give  an  accurate  de- 
scription of  all  the  phenomena  in  theii*  realm  of  experience,  but 
which  are  not  able  to  deduce  all  their  propositions  from  a  funda- 
mental set  of  propositions.  A  system  of  science  is  economical 
if  the  logical  processes  involved  ai'e  reduced  to  the  smallest  pos- 
sil)le  number.  The  quality  of  being  economical  involves  that 
the  set  of  fundamental  propositions  is  reduced  to  the  smallest 
possible  number;  i.  e.  to  those  propositions  which  are  necessary 
and  sufficient  to  deduce  all  the  propositions  of  the  system.  Prop- 
ositions of  this  type  are  independent  from  each  other;  this  quality 
can  be  proved  by  showing  that  not  all  the  propositions  of  the 
system  can  be  deduced,  if  one  of  the  fundamental  propositions, 
or  a  group  of  them,  is  omitted.  Every  proposition  of  a  set  of 
independent  propositions  may  be  substituted  by  its  logical  oppo- 
site without  the  conclusions  drawn  from  this  new  set  l)y  purely 
logical  processes  being  contradictory  among  themselves.  The  log- 
ical truth  of  a  system  of  science  is  the  consistence  of  any  one  of 
its  propositions,  or  of  any  group  of  propositions,  with  any  other 
proposition  of  the  same  system,  or  of  any  proposition  which  may 
be  derived  by  purely  logical  processes  from  the  same  set  of  fiui- 
damental  propositions.  A  set  of  fundamental  propositions  is  log- 
ically neither  true  nor  false'''  if  its  elements  are  independent  of 
each  other. 

*For  this  reason  they  are  sometimes  called  propositional  functions,  see  B. 
Russell,  Principles  of  Mathematics,  13,  353;  A.  N.  Whitehead,  Intro- 
duction logiquc  a  la  geometric,  Revue  de  Metaphysique  et  de  Morale,  \'ol.  15, 
1907,  p.  35. 


144  PROBLEMS  OP'  PSYCHOPHYSICS 

The  greatest  and  most  perfect  example  of  such  a  system  of 
science  is  geometry,  which  has  in  so  far  a  distinguished  position 
among  the  other  sciences  as  it  can  state  at  the  start  the  proposi- 
tions which  will  exhaust  its  field  of  work.  It  is  due  to  the  investi- 
gations into  the  foundations  of  geometry  that  one  has  acquired  an 
insight  into  the  relations  of  the  single  propositions  to  the  original 
suppositions  of  a  science,  since  it  was  seen  that  systems  of  consist- 
ent propositions  ma}' be  derived  from  fundamental  sets  of  proposi- 
tions in  which  one  proposition  is  superseded  by  its  opposite. 
Physical  sciences  have  at  present  not  yet  reached  this  high  state 
of  development,  but  there  is  little  doul)t  that  similar  sets  of  neces- 
sary and  sufficient  propositions  may  lie  constructed  for  the  differ- 
ent branches  of  mathematical  ph}-sics,  as  it  was  required  l:)y  Hil- 
bert.*  It  is  more  doubtful  whether  it  will  be  possible  to  construct 
similar  sets  for  the  biological  sciences,  because  one  has  not  yet 
found  principles  of  high  generality,  and  it  is  e.  g.  by  no  means  an 
easy  task  to  state  all  the  principles  which  are  common  to  phys- 
ical and  chemical  physiology.  The  fact  remains  that  one  tries 
to  construct  systems  which  cover  a  certain  range  of  the  field  of 
work. 

These  systems  have  nothing  to  do  with  the  existence  or  non- 
existence of  the  objects  they  speak  of.  A  system  of  science  be- 
comes applicable  to  a  certain  part  of  experience  only  by  means 
of  the  statement,  that  the  objects  contained  therein  are  such  as 
those  which  are  spoken  of  in  this  science.  This  application  to 
experience  is  the  test  of  the  empirical  truth  of  a  system  of  science. 
A  system  of  science  is  empirically  true  if  an  empirical  fact  corres- 
ponds to  every  one  of  its  propositions,  and  the  system  is  complete 
if  there  corresponds  to  every  empirical  fact  of  the  realm  of  expe- 
rience in  question  a  certain  proposition  of  the  system.  As  a  rule 
one  concludes  from  the  fact  that  certain  empirical  events  comply 
with  the  requirements  of  the  system  that  all  its  conclusions  must 
be.  true,'  but  one  also  can  argue  from  the  agreement  of  the  conse- 


**D.  HiLBERT,  Gdttinger  Nadir.  Math.-phys.  Kl.  1900,  p.  272  (also  Arch, 
f.  Math.  u.  Phys.  Vol.  i,  (3  ser.)  1901,  p.  62);  O.  HoeLDER  {.\nschauung  und 
Denkcn  in  der  Geometrie,  1900)  suspects  that  the  deductions  of  mechanics 
might  possibly  h^  not  as  pure  as  those  of  geometry. 


THE   PSYCHOMETRIC  FUNCTIONS  145 

quenccs  with  experience  to  the  truth  of  the  system.*  This 
logical  process  is  under  the  ,i:;eneral  principle  that  all  the  conclu- 
sions drawn  by  merely  logical  processes  from  a  proposition  which 
is  empirically  true  are  empirically  true.  This  may  be  illustrated 
by  the  example  of  geometry.  Those  ol)jects  which  we  call  spatial 
manifoldnesses  l)elong  to  three  different  types.  The  sets  of  fun- 
damental propositions  required  for  the  description  of  these  differ- 
ent manifoldnesses  are  identical  in,  all  but  one  proposition  and, 
since  this  proposition  is  independent  from  the  others,  one  may 
derive  different  systems  of  geometry,  in  which  every  proposition 
is  consistent  with  any  other  proposition,  or  group  of  propositions, 
of  the  same  system.  All  these  systems,  therefore,  are  logically 
true,  but  only  one  of  them  has  empirical  truth  too.  These  three 
manifoldnesses  differ  in  regard  to  a  certain  quantity,  the  so-called 
Gaussian  measure  of  curvature,  which  is  capable  of  empirical 
determination  and  the  question  which  of  the  three  systems  of 
geometry  is  empirically  true  is  reduced  to  the  qiiaestio  facti: 
Which  is  the  value  of  the  measure  of  curvature  of  empirical  space? 
Tho.se  measurements  which  show  that  physical  space  has  a  curva- 
ture which  can  differ  from  zero  only  b}^  a  quantity  which  is  smaller 
than  the  smallest  which  can  be  detected  by  our  instruments  of 
measurement,  is  the  empirical  warrant  that  all  the  conclusions  of 
Euclidean  geometry  are  empirically  true.  After  a  system  of 
science  is  established  and  its  empirical  basis  is  put  into  proper 
light,  it  is  possible  to  state  it  entirely  as  a  series  of  consequences 
of  a  simple  empirical  fact,  but  our  subjective  confidence  in  such 
systems  depends  materially  on  the  agreement  in  which  its  prop- 
ositions are  found  to  be  with    experience.     The  explanation  of  a 

*L.  BoLTZMAXN,  Uber  die  Entwicklung  der  Mcthoden  der  tlieoretischen  Physik 
in  neiiercr  Zeit,  Jahresbericht  d.  deutschen  Mathematiker-Vereinigung,  Vol. 
8,  1899,  p.  89,  says  that  our  confidence  in  the  fundamental  equations  of 
electrodj'namics  is  not  so  much  based  on  Ampere's  experiments,  which  allow 
us  to  state  them  as  on  the  general  agreement  of  experiment  with  all 
the  conclusions  which  may  be  drawn  from  them.  Tlie  case  of  geometry  as 
science  of  space  is  very  similar.  Nobody  doubts  geometry  but  not  because 
one  depends  so  much  on  those  facts  which  show  that  actual  space,  if  it  can  be 
regarded  as  a  geometrical  manifoldness,  is  a  Euclidean  manifoldness,  but 
because  all  the  propositions  of  geometry  have  been  found  to  correspond  to 
geometrical  facts  relating  to  einpirical  space 


146  PROBLEMS  OF  PSYCHOPHYSICS 

phenomenon  which  belongs  to  such  a  system  of  science  consists 
in  the  reduction  of  the  proposition  which  is  the  description  of 
this  phenomenon  to  the  system  of  fundamental  propositions  i.  e. 
in  showing  how  this  proposition  may  be  obtained  as  a  logical 
conclusion  from  them.* 

The  calculus  of  probabilities,  similarly,  is  a  system  of  proposi- 
tion derived  by, purely  logical  processes  from  the  notion  of  a 
mathematical  probability,  which  has  its  logical  foundation  in  the 
hypothetical  judgment.  There  exists  no  connection  between 
this  system  and  actual  events  and  one  tries  in  vain  to  deduce  any 
such  relation  by  means  of  the  theorem  of  Bernoulli  or  of  the  theo- 
rem of  Poisson.  These  propositions  become  applicable  to  a  cer- 
tain realm  of  experience  only  by  means  of  the  statement:  The 
events  of  this  realm  of  experience  have  the  character  of  ran- 
domness in  the  mathematical  sense  of  the  word.  It  follows  from 
this  statement  that  all  the  propositions  of  the  calculus  of  pro- 
babilities must  be  true  for  this  group  of  events,  but  the  truth  of 
every  single  proposition  is  based  and  derived  from  this  one 
empirical  fact  that  the  events  have  random  character.  The 
more  frequent  practice  of  concluding  from  the  agreement  of  all 
the  inferences  of  the  calculus  of  probabilities  with  experience  to 
the  randomness  of  the  events  stands  of  course  on  the  same 
epistemological  level. 

The  set  of  fundamental  proposition  may  be  subjected  to  criti- 
cism or  revision,  but  as  long  as  one  stands  on  the  ground  of  this 
science  one  has  no  possibility,  and  no  need,  to  doubt  the  truth 
of  the  set  of  fundamental  propositions.  All  one  can  do  is  to  state 
the  fact  that  the  conclusions  drawn  from  it  coincide  or  do  not 
coincide  with  experience.  It  is  not  always  possible  to  state  the 
fundamental  principles  in  such  a  way  that  one  can  be  sure  of  their 
empirical  truth;  this  is  the  case  mostly  when  the  facts  science  has 
to  work  on  are  yet  imperfectly  known.  In  these  cases  one  accepts 
the  formulation  which  fits  best  to  the  facts  known;  propositions 

*C.  F.  Gauss,  Magnetismus  und  M j-^ueto meter,  Werkc,  Vol.  5,  p.  315. 
"  Unter  Erklaren  versteht  aber  der  Naturforscher  nichts  anderes,  als  das 
Zuriickfiihren  auf  moglichst  wenige  und  moglichst  einfache  Grundgesetze, 
iiber  die  er  nicht  weiter  hinaus  kann,  sondern  sie  schlechthin  fordern  muss, 
und  aus  ihnen  die  Erscheinungen  vollstandig  ableitet." 


THE  P.SYt'HOMKTUIC  FINC'TIOXS  147 

of  I  his  kintl  are  hypotheses  of  generalizing  cliaracter.  Systems 
whicli  are  l)ase(l  on  generalizations  from  incomplete  data  are 
likely  to  undergo  changes  due  to  better  information.  The  cor- 
rections which  our  views  have  to  undergo  under  the  influence  of 
the  knowledge  of  new  facts  must  render  the  system  again  the  one 
that  fits  the  facts  best.  This  requirement  is  based  on  the 
methodological  principle  that  a  proposition  based  on  the  obsei'va- 
tion  of  empirical  facts  holds  goocU  for  experience  in  general  as 
long  as  no  contrary  instances  are  known.  The  system  of  science 
which  fits  best  the  facts  known  has,  therefore,  the  smallest 
probability  of  being  obliged  to  undergo  a  correction  under  the 
influence  of  new  information.  This  rule  may  l)e  formulated,  that 
one  chooses  among  systems  of  hypothetical  character  the  one 
which  has  the  greatest  stability. 

It  was  mentioned  above  that  the  difficulties  which  one  encoun- 
ters in  the  construction  of  these  systems  of  science  are  very  differ- 
ent in  the  difTerent  realms  of  experience.  The  totality  of  all  the 
data  which  constitute  experience  may  be  divided  into  two  classes. 
The  first  class  comprises  all  those  events  which  are  interpreted 
as  indicating  a  mental  life  similar  to  our  own;  the  second  class 
is  constituted  by  all  the  events  which  are  not  interpreted  in  this 
way.     Let  us  designate  events  of  the  fii'st  class  by  the  capital 

letters  A,   B,  C,  ,  events  of  the  second  class  by  the  small 

letters  a,  b,  c,  and  let  us  use  greek  letters  if  we  speak  of  an 

event  without  laying  stress  on  its  belonging  to  one  of  these  two 

classes.     The  description  of  the  phenomena  a,  b,  c,  has  made 

steady  progress,  but  that  of  the  phenomena  of  the  class  A,  B,  C, 

was  found  to  be  singularly  difficult.     The  elements  of  the 

class  a,  b,  c,  show  a  certain  constancy  in  their  arrangement 

which  makes  classification  comparatively  easy,  and  which  is  very 
important  for  the  study  of  the  succession  of  these  phenomena, 
understanding  by  succession  the  temporal  order  of  the  piienomena. 
This  succession  is  under  the  general  rule  of  the  law  of  causality 
which  states  that  there  are  certain  definite  rules  which  regulate 

the  order  in  which  a  group  of  events //,  v,  o,  ....  is  followed  by 

other  events  ....  /«',  v',  o' ,  If  the  rules  are  such  that  a  certain 

group  ....  p,  a,  r is  always  followed  by  the  same  group//,  a' ,-' , 

....,  we  may  say  that  the  succession  shows  uniformity,  but  if  the 


148  PROBLEMS  OF  PSYCHOPHYSICS 

rules  are  such  that  a  group  of  events  ....  p,  a,  z,  ....  may  be  fol- 
lowed at  different  times  by  different  gioups,  although  the  group 
which  follows  it,  is  well  defined,  we  may  say  that  the  succession 
is  regular.  The  law  of  causality  in  this  general  formulation 
does  not  discriminate  between  events  of  the  classes  A,  B,  C  .... 

and  a,  1),  c, The  difficulty  of  establishing  rules  of  succession 

depends  to  a  large  extent  on  the  possibility  of  identif3dng  groups 
of  events  and  on  their  stability,  which  consists  chiefl}'  in  their 
not  being  broken  up  or  interrupted  by  other  events.  The  num- 
ber of  rules  for  the  succession  of  the  events  of  the  class  a, 
b,  c,  ....  is  so  great  and  that  for  the  class  A,  B,  C,  ....  is  so  small, 
that  one  has  tried  to  utilize  this  fact  for  laying  down  a  general 
distinction  between  these  two  classes  of  events.  Leaving  the 
metaphysical  side  of  this  view  out  of  question  (that  the  class 
A,  B,  C,  ....  is  made  up  of  those  events  for  which.it  is  character- 
istic that  they  never  can  be  brought  under  the  law  of  causality) 
this  definition  is  certainly  not  serviceable  for  empirical  purposes, 
because  the  apparent  lack  of  rules  of  the  succession  is  also  found 
in  other  events  (events  of  random  character)  which  we  do  not 

classify  among  A,  B,  C,  Drawing  the  line  of  demarcation 

at  those  points  where  the  limits  of  our  actual  knowledge  are,  has 
the  inconvenience  that'the  class  A,  B,  C,  ....  is  cut  down  by  every 
progress  of  science.  The  logical  outcome  of  such  a  procedure 
is  the  denial  of  the  existence  of  this  class,  the  elements  of  which 
are  found  only  outside  of  the  realm  of  science,  a  denial,  which  in 
turn  is  confronted  with  the  fact  that  these  events,  whatever 
their  general  importance  may  be,  are  most  intimately  known  and 
of  greatest  immediate  interest  for  us. 

The  difference  between  these  two  classes  does  not  exist  for 
primitive  thinking,  which  is  anthropomorphic  to  such  an  extent 

that  all  the  events  seem  to  belong  to  the  class  A,  B,  C, The 

above  mentioned  elimination  of  this  class  belongs  only  to  a  com- 
paratively late  stage  of  development.  This  development  is  brought 
about  by  the  consistent  and  continuous  use  of  certain  notions  the 
origin  of  which  we  do  not  discuss  here  and  the  metaphysical  bear- 
ing of  which  we  leave  aside.  In  the  first  rank  among  these  no- 
tions stand  the  ideas  of  substance  and  causality.  Substance  is 
defined  as  that,  the  idea  of  which  is  the  absolute  subject  of  our 


THE     PSYCHOMETRIC    FUNCTIONS  140 

judgments  and  which  is,  therefore,  not  a  determination  of  any- 
thing. Substance  is  not  the  predicate  of  any  object,  but  it  is  the 
subject  to  which  all  attributes  refer.  All  phenomena  are  regarded 
as  attributes  of  something,  and  this  is  substance  which  remains 
unaltered  by  the  change  of  its  determinations.  The  difference  be- 
tween the  two  classes  of  events  being  established,  the  phenomena 
were  referred  to  two  entirely  different  substances,  and  the  problem 
was  to  define  them  and  to  answer  the  questions  which  arise  from 
the  application  of  these  notions  to  experience.  The  bearer  of  all 
the  predicates  of  the  class  a,  b,  c, ....  can  be  defined  as  what  is  move- 
able in  space,  and  the  term  used  for  this  notion  is  matter.  The 
substance  underlying  the  phenomena  A,  B,  C,  ....  is  characterized 
by  thinking.  These  are  the  characteristic  qualities  of  the  materia 
extensa  and  of  the  materia  cogitans,  from  which  their  other  deter- 
minations follow. 

These  notions  of  mind  and  matter  met  with  very  different  suc- 
cess. On  the  notion  of  matter  one  can  build  up  systemsof  almost 
inexhaustible  fertility,  but  all  the  efforts  to  deduce  anything 
from  the  notion  of  a  thinking  substance  have  been  in  vain.  In 
the  realm  of  experience  this  notion  is  of  no  use,  and  beyond  this 
it  leads  only  to  endless  controversies.  The  success  of  the  one 
notion,  however,  produced  a  favorable  prejudice  in  favor  of  the 
other  and  numberless  attempts  were  made  to  define  it  in  such 
a  way  that  the  difficulties  which  it  implies  may  be  avoided. 

We  may  call  every  theory  which  uses  the  notion  of  substance 
an  ontological  hypothesis,  because  it  expresses  a  view  on  the 
nature  of  things.  Any  ontological  theory  must  take  one  of  the 
following  views.  1.)  There  exists  only  one  substance,  because 
the  other  can  be  reduced  to  it;  2.)  there  exist  two  substances, 
one  of  which  produces  the  phenomena  A,  B,  C,  ....  and  the  other 

the  phenomena  a,  b,  c,  The  first  h3pothesis  is  the  monistic 

view,  which  may  be  formulated  as  materialism  or  as  spiritualism 
of  which  materialism  is  the  more  important  partly  for  reason  of 
the  consistency  of  the  system  and  partly  for  the  number  of  its 
followers.  Materialism  (spiritualism)  is  the  ontological  view 
that  extended  matter  (thinking  substance)  is  the  only  existing 
thing.  Both  forms  of  the  monistic  hypothesis  meet  with  one 
common  difficulty,  but  against  materialism  there  is  an  argument 


150  PROBI.EMS  OF  PSYCHOPHYSICS 

available  to  wliich  spiritualism  is  not  open.  This  argument  may 
be  called  the  epistemological  argument  against  materialism.  It 
is  based  on  the  fact,  that  what  is  immediately  given  to  us  is  con- 
sciousness and  that  matter  is  known  to  us  merely  by  our  ideas. 
To  reduce  mental  life  to  matter  is  therefore  an  explanation  per 
ignotius.-''- 

Every  monistic  system  is  called  upon  to  explain  how  a  substance 
may  produce  phenomena  so  fundamentally  different  in  their  qual- 
ities. Materialism  can  explain  everything  but  thought; spiritual- 
ism can  explain  everything  but  matter.  The  materia  cogitans  can 
not  be  reduced  to  the  materia  e.rtensa  nor  vice  versa.  Materialism 
and  spiritualism  are  equally  open  to  this  argument  which  may  l)e 
called  the  ontological  argument  against  monism.  This  argument 
is  greath^  used  in  the  different  refutations  of  materialism, f  find 
Leibnitz's  example  of  the  millj  is  very  likely  the  first  exhibition 
of  it.  No  new  argument  against  materialism  was  found  in  spite 
of  great  effort  and,  as  F.  A.  Lange  remarks, T[  it  is  always  the 
same  hit,  the  impossilnlity  to  reduce  the  psychic  to  the  physic, 
which  deals  a  knock-out  blow  to  materialism. 

*The  trend  of  this  argument  goes  back  to  K.vntt  K.  d.  r.  V.,  Werke,  ed. 
Hartenstein,  Vol.  3,  pp.  606,  607  (omitted  in  the  second  edition.)  The  full 
statement  of  this  subtle  argument  is  due  to  Schopenhauer,  IT.  a.  11'. 
u.  V.  Werke,  ed.  Griesebach,  Vol.  1,  p.  62  sqs.  "  Der  Materialismus  ist  also 
der  Versuch  das  unmittelbar  Gegebene  aus  dem  mittlebar  Gegebenem  zu 
erklaren."  This  argument  was  adopted  also  by  Riehl,  Schuppe,  Bergmann, 
Adickes  and  others;  see  L.  BussE,  Geist  und  K  or  per,  Seek  und  Leib,  1903, 
pp.  15-17,  who  states  the  argument  in  full  and  gives  references  to  further 
literature. 

tMaterialism  has  two  forms:  1.)  The  psychic  is  a  particular  kind  of 
matter;  2.)  the  psychic  is  a  type  of  motion.  The  latter  view  may  be  formu- 
lated more  cautiously  in  this  way:  The  effects  of  the  psychic  are  equivalent 
to  motion.  It  seems  that  this  formulation  avoids  the  ontological  argument 
but  in  this  formulation,  materialism  has  lost  entirely  its  ontological  charac- 
ter taking  the  jjlienomenalistic  point  of  view  that  a  force  is  known  to  us  merely 
by  its  efl'ects. 

JOne  finds  this  argun;:it  very  frequently.-  The  last  statement  is  that  of 
C.  L.  Herrick,  The  Nature  of  the  Soul,  Psych.  Rev.  Vol.  XIV,  1907,  i).  208, 
where  the  argument  is  attributed  to  Rabier. 

"IF.  A.  L.\NGE,  History  of  .Materialism  (Engl.  Trans!.,)  1881,  Vol.  Ill,  iJ. 
329.  For  the  critique  of  other  arguments  against  materialism  L.  BussE, 
Geist  und  Korper,  Sccic  und  Leib,  1903,  pp.  .50-61. 


THE  PSVCHOMKTUIC  FIXCTIONS  151 

Tlie  diialistic  Iwpotheses  are  in  turn  coufroutecl  with  the  neces- 
sity of  explaining  how  tw'O  substances  which  are  so  fundamentally 
different  can  influence  one  another.  The  relation  of  these  two 
substances  is  the  problem  to  be  explained  by  dualism;  the  intrin- 
sic difficulties  of  this  view  are  so  great  that  the  authors  who  pro- 
fess these  ideas  introduce  them,  very  frequently,  as  the  one  hy- 
pothesis against  which  the  smallest  number  of  arguments  tells.* 
8uch  a  view,  even  if  full  credit  is  given  to  all  the  arguments  of 
the  authors,  has  of  course  not  the  character  of  the  most  probable, 
but  only  of  the  least  objectionable  hypothesis,  and  it  amounts 
almost  to  a  renunciation  of  every  theoretical  intelligibility  to 
introduce  the  incomprehensible  right  at  the  beginning  of  one's 
explanation.!  The  dualistic  systems  occur  in  the  following 
three  forms:  1.)  Interactionalism,  2.)  occasionalism  and  3.) 
parallelism,  the  first  and  third  form  having  two  different  types. 

*Prof.  [AMES,  for  instance,  introduces  in  his  "Principles  of  Psychology" 
the  notion  of  a  soul  in  this  way  at  the  end  of  a  long  discussion  of  other  views; 
the  trend  of  Busse's  argumentation  is  similar. 

fThe  same  objection  holds  against  the  view  that  one  has  to  consider  it  as 
a  fact  that  the  psychic  influences  the  physical,  and  vice  versa,  and  that  the 
incomprehensibility  of  such  an  influence,  which  makes  this  influence  equiva- 
lent to  a  miracle,  is  no  objection,  "  because  a  miracle  that  happens  every  day 
ceases  to  be  a  miracle."  This  view  was  taken  by  Stumpf  and  Jerusalem. 
This  argument  proves  too  much.  It  is  for  instance  also  available  in  defence 
of  occasionalism,  which  is  thoroughly  acceptable  if  one  is  satisfied  with  resolv- 
ing the  events  in  an  uninterrupted  chain  of  miracles.  The  argument,  however, 
misses  the  following  point:  The  difficulty  of  the  problem  lies  in  the  correct 
definition  of  notions  which  must  be  serviceable  for  the  descri])tion  of  certain 
facts,  about  which  is  little  doubt  in  so  far  the  quaestio  facti  goes.  The  notions 
used  until  now  do  not  serve  this  purpose  of  a  correct  and  contradicticjnless 
description.  In  other  problems  one  would  dispose  of  notions  which  lead  to 
contradictions  and  try  new  ones.  If  the  notions  which  are  available  should 
not  meet  with  better  success  one  would  suspect  either  that  the  problem  is 
insolvable,  or  that  our  present  means  are  not  suificient  for  the  solution.  In 
the  first  case  one  is  satisfied  with  the  demonstration  that  the  problem  cannot 
be  solved  and  one  leaves  the  problem  aside,  as  it  was  done  with  the  i^roblems 
of  squaring  the  circle  and  of  constructing  the  perpetuum  mol)ile.  An  example 
of  the  second  possibility  is  the  problem  of  n  bodies,  o(  which  it  was  shown  lately 
that  our  present  means  are  not  sufllcient  for  a  general  solution.  The  pecuUarity 
of  the  mind-body  problem  lies  in  its  connection  with  other  highly  important 
questions,  which  make  it  desirable  to  solve  the  prol^lem  in  one  way  or  the  otiier. 


152  PROBLEMS  OF  PSYCHOPHYSICS 

Iiiteractionalism  is  found  as  the  theory  of  an  influxus  physicus 
or  as  the  theory  of  an  influxus  psychicus  or  as  a  combination  of 
these  two  theories.  The  first  type  of  the  third  form  of  the  dual- 
istic  hypothesis  is  the  pre-estabhshed  harmony, and  the  second 
is  psychophysical  parallelism  in  the  proper  sense,  the  principle 
of  which  is  that  mental  and  physical  phenomena  are  independent 
from  each  other,  so  that  one  of  them  cannot  be  reduced  to  the 
other,  and  that  they  go  parallel  without  being  in  causal  relation. 
Psychophysical  parallelism  was  stated  in  many  different  ways,, 
but  the  characteristic  feature  of  parallelistic  systems  is  the  alleged 
impossibility  to  reduce  psychical  to  physical  phenomena  or  rice 
versa  and  the  lack  of  causal  relations  between  them.  A  certain 
form  of  parallelism  (universal  parallelism  in  Busse's  terminology) 
approaches  closely  the  view  of  the  pre-established  harmony. 
Parallelism  is  called  upon  to  explain  the  intercourse  of  two  "spir- 
its,' '  as  e.  g.  in  conversation,  but  this  argument  avails  also  against  the 
other  dualistic  theories  except  interactionalism. 

The  dualistic  systems  meet  with  a  peculiar  difficulty  which  is 
caused  by  the  application  of  the  category  of  substance.  This 
difficulty  consists  in  the  impossibility  of  demonstrating  strictly 
the  existence  of  other  thinking  beings,  the  term  "thinking  being" 
referring  a  subject  which  is  the  bearer  of  conscious  states.  There 
is  no  necessity  for  the  assumption  of  conscious  beings  besides 
the  one  thinking  individual,  because  what  we  perceive  are  phe- 
nomena belonging  to  the  class  of  those  which  are  attributed  to 
the  materia  extcnsa,  and  only  our  own  conscious  states  are  imme- 
diately given  to  us.  This  view  is  known  under  the  name  of  ab- 
solute idealism  or  transcendental  egotism.  The  scholastic  for- 
mulation of  this  view  maintains  that  we  perceive  only  the  effects 
of  conscious  states  and  that  the  conclusion  from  the  effect  to  the 
cause  is  not  certain.  This  argument  seems  to  be  inevitable  un- 
der the  assumption  of  a  substance  which  underlies  the  thinking 
process  and  the  force  and  peculiarity  of  this  argument  is,  per- 
haps, best  characterized  b}^  Schopenhauer's  remark  that  trans- 
cendental egotism  remains  an  unconquerable  position  which,  how- 
ever, as  a  serious  point  of  view  can  be  found  only  in  the  mad- 
house. This  much  disputed  argument  seems  to  give  a  pecufiar 
position  to  the  mental  states  and  it  is  not  void  of  interest  that  it 


THE  PSYCHOMETRIC  FUNCTIONS  153 

may  be  applietl  to  objects  of  every  description.  This  Ijecomes 
obvious  at  once  when  a  question  is  raised  like  this:  How  do  we 
know  that  there  is  magnetic  substance  in  a  magnet?  A  magnet 
is  defined  by  the  qualities  of  bodies  in  general  and  in  addition  to 
these  by  all  those  reactions  which  are  characteristic  for  the  mag- 
netic state.  There  is  nothing  about  a  substance  in  the  data  and 
all  we  know  about  a  magnetic  body  is  exhausted  with  their  de- 
scription. In  the  same  way  we  da  not  know  of  intelligence  in  a 
thinking  individual  except  by  its  reactions  which,  however,  are 
not  as  clearly  defined  as  those  characteristic  for  magnetism,  and 
which  are,  furthermore,  less  stable.  This  lack  of  stability  is  the 
reason  why  the  notion  of  substance  is  of  so  little  use  in  the  treat- 
ment of  psychical  phenomena  and,  no  matter  whether  we  regard 
with  Mach  the  notion  of  a  substance  as  a  hypothesis  introduced 
for  the  sake  of  explanation,  or  whether  we  consider  it  with  Kant 
as  one  of  those  notions  which  we  are  bound  to  apply  to  experi- 
ence— two  views  which  are  by  no  means  so  very  widely  different 
and  which  may  be  varied  in  detail  considerably — the  fact  remains 
that  the  notion  of  substance,  though  very  useful  in  physics,  is 
of  no  practical  avail  in  psychology.  In  physics  the  notion  of 
substance  leads  to  that  of  matter,  as  that  which  is  extended, 
moveable  and  impermeable,  but  when  appUed  to  mental  states 
it  can  be  used  only  as  the  notion  of  an  indefinite  "something"^ 
which  is  of  no  use.  It  is  a  very  obvious  idea  to  try  how  much  one 
can  do  without  this  notion. 

In  order  to  avoid  all  these  diflficulties  one  may  try  to  formulate 
the  problem  in  such  a  way  that  it  loses  its  ontological  character, 
retaining  only  the  general  problem  of  finding  relations  between- 
phenomena.  Events  are  the  only  immediate  datum  of  experi- 
ence and  one  may  try  to  find  relations  between  them.  The 
complete  description  of  an  event  requires  that  all  the  events 
which  are  connected  with  it  according  to  a  definite  rule  are  de- 
scribed. A  mental  .state  is  connected  not  only  with  events  of 
the  class  A,  B,  C,  ....  but  also  with  events  of  the  class  a,  b,  c,  .... 
which  must  not  be  omitted  in  a  complete  description.  Psycho- 
physical parallelism,  thus,  gets  a  very  simple  expression  on  tliis 
ground.  The  general  principle  of  psychophysical  parallelism 
is  that  the  events  A,  B,  C,  ....  are  in  a  relation  with  the  events 


154  PROBLEMS  aF  PSYCHOPHYSICS 

a,  b,  c,  ....  and  that  a  mental  phenomenon  A  is  not  explained 
before  those  events  of  the  class  a,  b,  c,  ....  are  described  with  which 
A  is  in  a  definite  relation.  This  principle  does  not  make  any 
definite  assumption  about  the  nature  of  this  relation,  but  it  is 
rather  the  exjjression  of  the  fact  that  the  phenomena  of  the  classes 
A,  B,  C,  ....  and  a,  b,  c,  ....  constitute  together  the  realm  of  experi- 
ence, in  which  we  ma}'  look  out  everywhere  for  connections  be- 
tween the  events.  A  practical  consecjuence  of  this  principle  is  that 
the  investigation  of  the  regularities  of  the  connection  between 
events,  especially  of  events  of  the  class  A,  B,  C,  ....,must  not  stop 
at  any  boundary  inside  the  realm  of  experience.  The  existence 
of  definite  rules  for  the  connection  of  events  is  a  supposition  of 
the  possibility  of  constructing  scientific  systems,  because  it  would 
not  be  possible  otherwise  to  deduce  an  exhaustive  system  from  a 
finite  number  of  principles  The  supposition  that  there  are  no  dis- 
connected events,  i.  e.  wonders,  in  the  entire  realm  of  experience  is 
the  condition  of  a  complete  description  of  the  phenomena.  In  the 
description  of  an  element  of  the  class  A,  B,  C,  ....  ,  e.  g.  a  color  sen- 
sation, the  problem  of  psychophysical  parallelism  is  not  exhausted 
with  the  answer  to  the  cjuestion:  Which  are  the  phenomena 
of  the  class  a,  b,  c,  .  ..  which  are  connected  with  this  event  accord- 
ing to  a  definite  rule?  The  class  of  phenomena  the  description 
of  which  is  required  by  this  question  contains  among  other  pro- 
cesses those  which  take  place  in  the  optic  nerves  and  in  the  occip- 
ital lobes  under  the  influence  of  retinal  stimulation.  This  answer 
shows  in  itself  that  this  description  is  only  a  part  of  the  answer 
to  the  more  general  problem  which  does  not  restrict  the  events 
to  those  of  the  second  class,  and  which  requires  the  description 
of  all  the  events  with  which  the  first  event  is  in  relation.  Events 
■of  the  class  A,  B,  C,  ....  have  the  same  claim  as  those  of  the  class 
a,  b,  c,  .  ..  and  the  view  that  psychology,!,  e.  the  science  which 
deals  primarily  with  the  events  A,  B,  C,  ....,  should  be  resolved 
into  a  special  science  of  the  events  a,  b,  c,  ....,e.  g.  into  brain  phys- 
iology, is  not  essentially  superior  to  the  opposite  view  that  the 
events  a,  b,  c,  ....  are  indifferent  for  psychology,  although  it  may 
lead  at  present  to  a  greater  number  of  propositions,  because  the 
relations  of  the  second  class  are  better  known.  The  statement 
that  mental  phenomena  depend  on  physical  phenomena  is  fre- 


THE   PSVCH(3.METRIC  FUNCTIONS  155 

quently  nothinji  else  but  an  expression  of  the  principle  of  psycho- 
physical parallelism.  Sometimes,  however,  this  statement  is 
ijiven  the  more  definite  meaning  that  spatially  well  defined  groups 
of  events  of  the  class  a,  b,  c,  ..  .  are  in  constant  relation  with 
certain  events  of  the  class  A,  B,  C,  ....  so  that  the  event  A  does 
not  occur  if  ....m,  n,  o,  ....  did  not  occur,  and  that  ....  m,  n,  o,  .... 
occur  when  A  is  observed.  Such  a  statement  contains  a  special 
law  and  is  necessaril}^  the  product  of  observation.  The  best 
known  example  of  a  relation  of  this  type  is  the  so-called  principle 
of  cortical  localization.  The  term  "principle"  is  well  chosen  if 
this  proposition  is  given  the  meaning  that  there  exist  such  groups 

of  events  ....  m,  n,  o,  ....  for  every  event  of  the  class  A,  B,  C, 

This  proposition  is  of  higher  generality  than  a  result  of  observa- 
tions can  l3e  and  it  deserves  the  name  of  a  principle,  because  it  lays 
down  a  rule  for  an  important  part  of  physiology.  The  term  "prin- 
ciple of  cortical  localization,"  however,  is  not  well  chosen,  if  it  is 
applied  to  that  group  of  propositions  which  are  the  outcome  of  all 
the  investigations  along  this  line.  This  group  of  propositions  is  the 
expression  of  empirical  facts  and  the  term  "empirical  law"  or 
"law"  is  more  appropriate.  This  relation  between  events  A, 
B,  C,  ....  and  events  of  the  class  a,  b,  c,  ....  seems  to  be  very  mys- 
terious, if  this  relation  is  conceived  as  a  relation  between  two 
fundamentally  different  substances,  but  this  mystery  disappears 
if  the  purely  phenomenological  point  of  view  is  taken.  Both 
events  stand  on  the  same  level  and  their  relation  is  not  more 
mysterious  than  that  between  any  two  others.  The  existence 
of  definite  relations  between  spatially  well  defined  groups  of 
physical  phenomena  and  certain  mental  phenomena  shows  that 
it  is  l)y  no  means  an  impossible  task,  involving  a  contradiction,  to 
establish  relations  between  phenomena  of  the  class  A,  B,  C,  ....  and 

those  of  the  class  a,  b,  c, *     The  contradiction  only  comes  in 

w'hen  the  phenomena  are  referred  to  substances  which  are  funda- 
mentally different.     Treating  the  data  of  experience  in  a  purely 

*.\n  analysis  of  a  very  complicated  group  of  sensation  into  its  elements 
was  given  by  E.  M.^ch,  Analyse  dcr  Etnpfinduiigcn,  1902,  pp.  32,  sq.  This 
author  describes  in  this  book  as  well  as  in  his  "  Thermodynamics"  and  in  his 
"  .Mechanics"  the  motives  which  may  induce  us  to  give  to  the  objects  a  reality 
independent  of  our  sensations. 


156  PROBLEMS  OF  PSYCHOPHYSICS 

phenomenological  way  without  making  use  of  the  idea  of  sub- 
stance shows  that  the  barrier  which  separates  the  class  A,  B,  C, 
....  from  the  class  a,  b,  c,  ....  it  not  insurmountable,  or  rather 
that  the  problem  of  establishing  relations  between  the  events  of 
these  two  classes  is  not  essentially  different  from  the  problem  of 
finding  relations  between  other  groups  of  phenomena. 

The  inadequacy  of  the  substitution  of  relations  l)etween  sub- 
stances for  the  purely  phenomenological  relations  between  events 
is  most  apparent  in  the  problem  of  the  relation  of  the  psychical 
to  the  phj'sical.  The  view  that  all  events  are  modifications  of 
substances  may  be  called  the  theory  of  mechanical  or  substantial 
causality;  this  view  implies  that  substances  are  the  ultimate 
reality  of  everything.  The  shortest  expression  of  this  view  that 
substance  is  the  factor  undergoing  changes  is  given  in  the  sentence: 
All  causing  is  effecting  (Alles  Wirken  ist  ein  Bewirken).'^=  This 
view  is  eminently  anthropomorphic,  because  it  takes  the  will  act 
as  the  type  of  a  cause.  The  difficulty  of  conceiving  of  the  rela- 
tion between  thinking  substance  and  extended  substance  is  by  no 
means  the  only  shortcoming  of  the  notion  of  substantial  causality. 
This  view,  in  fact,  leads  to  many  other  difficulties  and  contradic- 
tions as  e.  g.  to  the  statement  that  every  judgment  must  express  a 
relation  between  substances,  or  between  a  sulDstance  and  its 
accidentibus.  The  only  type  of  judgments  admissible  under  this 
supposition  are  those  judgments  in  which  the  relation  of  a  sul)- 
ject  to  its  predicates  are  determined.  This  classification,  how- 
ever, does  not  provide  for  the  so-called  impersonal  judgments 
to  which  this  definition  does  not  apply.  This  fact  shows  that 
the  notion  of  substantial  causality  does  not  exhaust  even  those 
phenomena  which  have  no  direct  bearing  on  the  mind-body  prob- 
lem. It  is,  therefore,  advisable,  to  see  whether  the  notion  of 
substantial  causality  cannot  be  superseded  ,by  another.  We  shall 
use  the  notion  of  relations  between  phenomena;  causality,  then, 
is  conceived  as  a  type  of  order  between  events. 

*The  exponents  of  this  theory  are  fairly  numberless,  and  we  cjuote  only  as  an 
example  C.  A.  Strong,  117;;'  the  Mind  has  a  Body,  1903,  p.  75,  "A  voHtion 
seems  the  very  type  of  a  cause."  The  most  recent  exposition  of  this  view 
was  given  by  Professor  Frank  Thilly  in  his  article  " Caiisality,"  Philosoph- 
ical Review,  Vol.  16,  1907. 


THE    PSYCHOMETRIC     FUNCTIONS  157 

The  term  relation  in  its  broadest  signification  has  the  meaning 
of  a  one-to-one  relation  between  two  elements;  this  notion  if  it 
is  confined  to  ciuantities  coincides  with  the  notion  of  a  mathemat- 
ical function  in  the  sense  of  Cauchy  and  Dirichlet.  Such  a  func- 
tion is  a  rule  by  which  a  value  of  the  dependent  variable  is  ad- 
joined to  every  admissible  value  of  the  independent  variable. 
The  most  general  form  of  a  relation  is  that  of  a  non-uniform, 
discontinuous  function.  This  definition  is  too  general  to  be 
treated  by  our  present  means  with  advantage,  and  one  restricts 
the  investigations  first  to  the  uniform,  continuous  functions,  and 
later  on  it  becomes  necessary  to  restrict  it  still  further  to  the 
analytic  functions.  A  continuous  function  may  be  given  by  a 
numerable,  infinite  set  of  conditions,  but  this  is  not  the  case  of 
the  general  discontinuous  functions.  These  functions  are  char- 
acterized by  an  innumerable,  infinite  set  of  conditions,  what  is 
the  same  as  to  say  that  they  cannot  be  defined.  In  consideration 
of  the  fact  that  Fourier's  analysis  shows,  how  certain  discontin- 
uous functions  may  be  represented  by  a  sum  of  continuous 
functions  one  might  have  taken  the  view  that  these  general 
functions  may  be  represented,  or  approximated,  by  analytic 
functions.  This  view,  however,  would  hardly  have  been  a  rea- 
sonable, expectation,  and  it  was  made  impossible  lately  by  the 
discovery  of  Baire,  that  every  function  which  can  be  represented 
as  the  limit  of  continuous  functions  must  be  a  function  with  not 
more  than  punctual  discontinuities.  If  it  should  happen  that 
a  function  of  more  than  punctual  discontinuities  occurs  in  na- 
ture, it  would  be  not  only  impossible  to  represent  it  in  the  usual 
way  but  it  even  would  be  impossible  to  approximate  it.  An 
event  of  this  type  could  not  be  described  in  the  same  terms  as 
ordinary  events.  A  classification  of  events  of  this  type  cannot 
lie  exhaustive  and  it  is  not  possible  to  deduce  systems  of  science 
from  a  limited  number  of  propositions,  because  there  is  no  war- 
rant that  one  may  not  find  at  any  moment  an  event  which  does 
not   reseml)le  in  anything  any  element  of  previous  experience. 

*See  BoREL,  L^qons'  sur  la  theorie  de^  fonctions;  Lbbesgue,  Leqons  sur 
r integration;  B.^IRE,  Les  fonciions  discontinues;  Philip  E.  B.  Joi-rdain 
The  Development  of  the  Theory  of  Transfinite  Numbers,  Archiv  der  Mathematik 
und  Pkysik,  Ser.  III.  Vol.  10,  1905,  pp.  2.54-281. 


158  PROBLEMS  OF  PSYCHOPHYSICS 

It  may  be  a  question  under  which  conditions  systems  of  science 
are  deducible  from  a  limited  set  of  propositions,  using  only  a 
limited  number  of  fundamental  ideas  or  notions,  but  there  is  no 
doubt  that  such  systems  cannot  possibly  be  exhaustive  of  a  cer- 
tain realm  of  experience  if  non-uniform,  discontinuous  functions 
are  admitted.  Winter*  has  tried  to  argue  against  the  possil^ility 
of  deriving  a  system  of  science  from  a  limited  number  of  funda- 
mental notions  by  means  of  the  fact  that  mathematics  in  its  nat- 
ural progress  constantly  introduces  new  notions,  the  definition  of 
which  is  apparently  more  or  less  arbitrary.  Such  is  e.  g.  the 
origin  of  the  elliptic  functions  which  are  obtained  by  increasing 
the  order  of  a  function  in  a  certain  integral  which  leads  to  known 
functions  by  the  unit.  Against  this  argumentation  one  may 
say  that  the  way  pointed  out  by  Winter  is  the  way  of  the  discov- 
ery of  new^  functions,  but  that  every  function  can  be  derived  by 
starting  from  the  series  which  gives  the  definition  of  the  function. 
This  series  is  constituted  by  no  other  l)ut  known  operations,  so 
that  the  new  function  is  expressible  by  an  algorithm  which  con- 
tains no  other  but  elementar}'-  operations  of  which  there  is  only 
a  limited  number.  The  introduction  of  new  functions,  therefore, 
is  not  an  argument  against  the  possibility  of  a  >ystem  being  built 
up  on  a  limited  number  of  notions,  as  long  as  the  new  functions 
can  l^e  reduced  to  the  forms  of  relations  already  known.  This 
possibility  is  not  given  for  functions  which  belong  to  the  general 
type  of  a  function  in  the  sense  of  Cauchy  and  of  Dirichlet. 

We  have  introtluced  above  the  cUstinction  between  regularity 
and  unifoi'mity  of  events.  Uniformity  is  the  more  special  case 
of  regularity,  because  it  may  be  that  the  rules  of  the  succession 
of  phenomena  vary  in  such  a  way  that  different  rules  of  succession 
are  adjoined  to  the  same  phenomenon  at  different  times.  The 
type  of  order  which  is  applied  to  the  study  of  natural  phenomena 
is  the  one  of  exclusive  uniformity,  requiring  that  the  same  group 
of  events  under  given  conditions  is  alwaj's  followed  by  a  certain 
other  group.  These  relations  are  most  commonly  found  in  mechan- 
ics, and  for  this  reason  one  frequently  calls  this  type  of  relation 

*M.  Winter,  Suy  I' introduction  logique  a  la  theoric  dcs  fonctioiis,  Revue  de 
Meta physique  et  de  Morale,  Vol.  15,  1907,  pp.  205-209. 


THE  PSYCHOMETRIC  FUNCTION'S  lo!) 

mechanical  causality.  The  simplest  example  of  events  of  this 
kind  is  the  push  and  pull  of  one  mass  on  another  mass,  and  the 
task  of  constructing;  a  system  of  science  in  which  the  events  are 
causally  determined  was  frequently  formulated  in  this  way,  that 
every  event  must  be  analyzed  into  motions  of  masses  which  are 
moved  by  the  push  and  pull  of  other  masses  and  which  have  no 
power  of  their  own  to  influence  their  motion  {vis  a  tergo).'^  The 
beauty  of  this  conception  is  that  the  influence  of  one  mass  on  the 
motion  of  another  mass  seems  to  be  immediately  intelligible  by 
the  analogy  of  the  influence  of  our  body  on  its  surroundings. 
The  consideration  of  mechanical  phenomena,  however,  leads 
only  to  one  class  of  relations,  and  it  was  seen  at  a  very  late  time 
that  the  problem  thus  restricted  excluded  all  relations  which  do 
not  belong  to  a  comparatively  small  class.  The  class  of  relations 
which  are  admitted  for  the  description  of  natural  phenomena 
are  the  so-called  analytic  functions.  The  view  that  all  phenomena 
are  under  the  principle  of  mechanical  causality  requires  that 
natural  phenomena  are  described  merely  in  terms  of  analytic 
functions.  These  functions  have,  besides  some  other  minor 
properties,  the  following  distinguishing  peculiarities:  They 
are  single  valued,  continuous,  differentiate,  they  can  be  developed 
into  a  power  series,  they  admit  of  an  analytic  continuation,  and 
they  are  solutions  of  differential  equations.  These  peculiarities 
make  the  analytic  functions  so  precious  for  the  study  of  natural 
phenomena.  The  conditions  in  infinitesimally  small  intervals 
of  time  and  space  are  so  simple,  that  we  can  express  them  with 
comparative  ease  by  differential  equations,  the  integration  of 
which  leads  necessarily  to  analytic  functions.  Having  established 
such  a  relation  in  no  matter  how  small  an  interval  of  time  and 
space,  we  can  determine  and  foresee  every  phase  of  the  later  de- 

*The  value  of  this  view  is  the  question  at  issue  in  the  discussions  between 
the  adherents  of  the  energetic  view  and  those  of  the  atomistic  hypothesis. 
The  denial  of  the  indispensability  and  of  the  adequacy  of  the  atomistic  hy- 
pothesis is  the  contention  of  the  energists  (E.  Mach,  Die  Mcchanik  m  I'mcr 
Entuicklung,  p.  486  "  Dass  alle  Vorgange  mechanisch  zu  erklaren  sind  halten 
wir  fiir  ein  Vorurtheil,")  whereas  the  adherents  of  the  atomistic  hypothesis 
point  out  that  the  energetic  view  does  not  show  a  success  in  the  explanation 
of  natural  phenomena  nearly  equal  to  that  of  the  atomistic  hypothesis. 


160  PROBLEMS  OF  PSYCHOPHYSICS 

velopment  of  the  process,  because  the  function  which  describes 
this  process  has  an  analytic  continuation.  The  course  of  events, 
which  is  the  course  of  nature,  is  uniquely  determined  so  far  as  it 
is  characterized  by  analytic  functions.  Events  which  are  char- 
acterized by  analytic  functions  are  completely  under  our  control 
and  none  of  the  details  of  such  events  can  escape  detection. 
Physics  deals  exclusively  with  processes  which  may  be  described 
by  analytic  functions  and  the  success  of  this  science  sug.ejested 
the  view  that  all  events,  if  properly  analyzed,  must  lead  to  this 
type  of  relations.*  If  it  is  true  that  there  are  in  nature  no  other 
but  analytic  functions,  then  there  exists  indeed  the  possibility  of 
exhausting  the  description  of  the  world  by  the  future  progress 
of  science.  The  observation  of  an  event  in  no  matter  how  small 
an  interval  of  time  and  of  space  permits  to  predict  the  future  and 

*We  quote  only  B.  Riemann,  Uber  die  Darstellbarkeit  einer  Function  durch 
eine  trigonometrische  Reilie,  M'erke,  ed.  Weber,  1892,  p.  237:  "Durch  die 
Arbeit   Dirichlet's  ward  einer  grossen   Menge  wichtiger  analytischer  Unter- 

suchungen  eine  feste  Grundlage  gegeben In  der  That  fiir  alle  Falle  der 

Natur,  um  welche  es  sich  allein  handelt,  war  sie  vollkommen  erledigt,  denn 
so  gross  auch  unsere  Unvvissenheit  dariiber  ist,  wie  sich  die  Krafte  und  Zu- 
stande  der  Materie  im  UnendUchkleinen  andern,  so  konnen  wir  doch  sicher 
annehmen,  dass  die  Functionen,  auf  die  sich  Dirichlet's  Untersuchung  nicht 
erstreckt,' in  der  Natur  nicht  vorkommen."  The  functions  which  Diriclilet's 
investigation  does  not  cover  are  the  functions  with  an  infinite  number  of 
maxima  and  minima  in  a  finite  interval,  and  the  supposition  that  there  are 
in  nature  no  other  but  analytic  function  comes  out  clearly  in  Riemann's  words. 
The  impossibility  of  the  occurrence  of  functions  with  an  infinite  number  of 
maxima  and  minima  in  the  description  of  the  motion  of  a  point  can  be  seen 
in  this  way.  A  body  which  moves  along  a  curve  can  approach  a  point  in  the 
neighborhood  of  which  there  is  an  infinity  of  maxima  and  minima  only  with 
always  decreasing  velocity,  and  a  movement  from  this  point  is  impossible 
(See  A.  KoEPKE,  Difjerentirbarkeit  und  AnschaulichkeH  von  Functionen. 
Math.  Annalen,  Vol.  29,  1887,  pp.  137-140.)  The  reason  for  the  requirement 
of  a  difi'erentiable  function  is  that  a  motion  which  is  characterized  by  a  func- 
tion without  a  derivative  would  have  to  take  place  without  a  definite  velocity 
or  acceleration.  Some  of  the  modern  writers  have  introduced  diilerent labil- 
ity as  a  specific  requirement,  e.  g.  Heumholtz,  V orlcsiingen  iiber  die  Dyna- 
mik  diskreter  Massenpunkte  {Vorlesungen  nber  theorctische  Physik,  Vol.  1,  1898,) 
p.  7,  sq.;  L.  BoLTZMANN,  Vorlesungen  iiber  die  Prinzipe  der  Mechanik,  1897, 
pp.  10-13,  who  says  that  differentiability  ought  to  be  introduced  as  one  of  the 
requirements  of  mechanics. 


I 


k 


THE  PSVCHOMETKIC  FUNCTIONS  IGl 

to  State  every  event  of  the  past.  The  events  at  any  particuhir 
point  of  time  and  space  depend  in  a  very  complicated  but  perfectly 
definite  way  on  all  the  other  events,  and  it  is  merely  a  question  of 
ability  how  much  of  the  world's  course  in  general  one  may  be  able 
to  deduce  from  one  single  fact  no  matter  how  trivial  it  may  be. 
The  perfect  knowledge  of  all  these  relation  would  be  the  realisa- 
tion of  Laplace's  mechanical  ideal,  which  is  only  a  consequence 
of  the  supposition  of  the  exclusive  occurrence  of  analytic  functions 
in  nature  i.  e.  of  the  absolute  intelligibility  of  all  natural  phenomena. 
►Special  mechanical  principles,  as  e.  g.  the  theorem  of  the  conser- 
vation of  energy,  depend  on  the  principle  of  general  causality 
formulated  as  the  exclusive  occurrence  of  analytic  functions  in 
nature,  and  they  do  not  hold  generally  if  other  functions  are 
admitted. '■■'■  This  view  that  all  events  are  characterized  by  ana- 
lytic functions  and  that  ever^'thing  can  be  treated  by  the  same 
scientific  method  was  called  materialism  by  Alexejef¥,t  but 
since  this  term  is  used  for  so  many  widely  different  notions  and 

*As  long  as  no  sets  of  necessary  and  sufficient  axioms  of  mechanics  are 
CLinstructed  it  is  very  hard  to  tell  whether  a  proposition  has  the  character 
of  a  principle  or  of  a  law.  Sigw.\rt  {Logik  2,  ed.  1893  \'ol.  2,  p.  644)  says 
he  cannot  convince  himself  that  Galilei's  inertia  and  Newton's  universal  at- 
traction are  necessary  consequences  of  the  principles,  so  that  he  would  be 
inclined  to  regard  them  as  natural  laws.  This  may  be  true  for  the  special 
form  of  the  Newtonian  potential  but  it  is  not  equally  obvious  for  the  law  of 
inertia  which  was  frequently  formulated  as  a  negative  expression  of  the  prin- 
ciple of  causality.  For  literature  on  the  problem  of  inertia  see  L.  Lang 
Das  Inertialsystem  vor  clem  Forum  der  Nafuriiissenscliajt,  Phil.  StiuL  Vol.  20 

1902,  pp.  1-71 .  The  proposition  of  the  conservation  of  the  energy  w-as  regard- 
ed by  Hemholtz  as  a  consequence  of  the  causalistic  conception  of  the  w'orld, 
whereas  other  investigators  try  not  very  successfully  to  look  at  it  as  a  datum 
of  experience,  e.  g.  Erich  Becher,  Das  Gesefz  von  der  Erhaltung,  der  Energie 
efc'  Ztschr.  f.  Psychologie,  Yo\.  46,  1907,  p.  98.  On  the  principle  of  the  con- 
servation of  energy  see  W.  Wundt,  Physiologische  Psychologic,  5,  ed.  Vol.  '^^, 

1903,  p.  634.  A  discussion  of  the  views  on  this  question  as  advocated  by 
different  authors  is  given  by  L.  BussE,  Geist  und  K  or  per,  Secle  und  Leib,  1903, 
p.  119,  124-126. 

fW.  G.  AlEXEJEFF,  Die  arithmologische  und  ivahrscheinlichkeitstheore- 
tische  Kausalitdt  etc.,  Ztschr.  /.  Philosophie  u.  Pddagogik,  Vol.  14,  Nov.  1905. 
pp.  50-55  speaks  of  "Materialismus  oder  die  rein  physikalische  Betrachtungs- 
weise  des  Geistigen."  We  use  only  the  second  term  of  the  two  suggested  by 
Alexejeff. 


102  PROBLEMS  OF  PSYCHOPHYSICS 

since  it  is  primarily  a  term  for  an  ontological  view,  it  may  seem 
best  to  find  another  name  for  it.  The  term  "mechanistic  view" 
would  be  appropriate,  if  it  were  not  frequently  used  for  the  desig- 
nation of  the  atomistic  hypothesis,  and  it  is,  perhaps,  best  to  call 
this  view  after  physics,  the  science  which  most  successfully  uses 
it.  By  the  term  "physical  point  of  view"  we  intend  to  designate 
the  view  that  there  are  no  other  but  analytic  functions  to  be  found 
in  nature.'^' 

The  problem  of  the  description  of  mental  life  calls  for  the  es- 
tablishment of  the  laws  of  the  succession  of  these  phenomena 
and  the  broadest  solution  of  this  problem  would  be  to  find  a  one- 
to-one  relation  between  every  moment  of  time  and  a  certain 
element  of  the  class  A,  B,  C,  ....  ,  but  leaving  the  question  open 
whether  this  relation  has  the  character  of  an  analytic  function 
or  not.  The  way  of  proceeding  may  be  illustrated  by  the  fol- 
lowing example  taken  from  physical  science.  The  temperature 
of  a  liquid  can  be  raised  by  bringing  it  into  contact  with  a  body 
of  higher  temperature.  The  temperatures  of  the  body  and  of  the 
liquid  vary  until  the  flow  of  heat  comes  to  rest  at  a  certain  time, 
but  in  every  moment  the  liquid  has  a  certain  temperature  as  well 
as  the  body,  so  that  there  is  a  one-to-one  relation  l^etAveen  the 
moments  of  time  and  the  temperatures  of  the  liquid.  The  ac- 
curate description  of  this  process  is  supplied  by  the  mathematical 
relation  which  shows  how  the  temperature  of  the  liquid  depends 
on  time.  A  similar  example  of  psychology  would  be  found  in 
the  description  of  the  succession  of  the  mental  states  under  the 

*It  happens  not  unfrequently  that  the  supposition  that  an  event  must 
have  an  analytic  expression  is  superseded  by  the  other  supposition  that  it 
must  be  a  simple  expression.  An  example  may  be  found  in  B.  A.  GotiLD, 
Use  of  the  Sine-formula  for  the  Diurnal  Variations  of  Temperature.  Amer.  Jl 
of  Science,  Vol.  119,  1880,  who  tries  to  show  (p.  217)  the  importance  of  finding 
general  expressions,  because  an  expression  which  can  represent  n  observa- 
tions but  depends  on  a  number  of  parameters  smaller  than  n  is  a  real  natural 
law.  A  discussion  of  tliis  view  may  be  found  in  H.  Burkh.\rdt,  Entwick- 
lungen  nach  oscillierenden  Reihcn,  Jahreshericht  d.  deuischen  Mathematiker 
Vereinigung,  1902,  pp.  249,  250.  Burkhardt's  opinion  that  this  view  is  char- 
acteristic of  the  American  for  Anglo-American  conception  of  natural  science  is 
perhaps  not  entirely  justified.  The  conception  that  a  natural  law  must  be  a 
simple  law  leads  of  course  to  a  very  flat  rationalism. 


THE    PSYCHOMETRIC     FUNCTIONS  163 

infiuonce  of  an  incoming  stimulus;  the  complete  solution  of  tins 
proljlem  would  give  a  one-to-one  relation  between  every  moment 
of  the  time  under  consideration  and  a  certain  mental  content. 
There  is  no  doubt  about  it  that  mental  phenomena  are  functions 
of  time  and  the  difficulty  merely  consists  in  defining  these  func- 
tions. We  dismiss  from  the  start  the  supposition  that  the  nature 
of  this  relation  is  known  before  hand.  It  seems  to  be  very  obvious 
to  draw  the  following  erroneous  conclusion. 

There  exists  a  one-to-one  relation  between  the  elements  A,  B, 
C,  ....  and  the  moments  of  time — no  specific  hypothesis  being 
made  about  the  nature  of  this  relation — but  there  exists  also  a 
definite  relation  between  the  moments  of  time  and  the  elements 

a,  b,  c,  One  supposes  that  the  latter  relation  must  belong 

to  the  causal  type  i.  e.  that  it  must  be  expressible  in  analytic 
functions.  Instead  of  merely  concluding  that  there  must  be  a 
relation  of  some  kind  between  the  elements  A,  B,  C,  ....  and  the 
elements  a,  b,  c,  ....,  one  may  try  to  specialize  these  relations  to  the 
type  of  analytic  functions.  This  conclusion  is  of  course  unwar- 
ranted, because  the  combination  of  an  analytic  function  with  a 
non-analytic  function  does  not  give  an  analytic  function.  We 
may  illustrate  the  far  reaching  importance  of  this  supposition 
by  Mr.  Montague's  theory  of  the  specious  present.*  This  author 
starts  from  the  observation  that  physical  and  psychical  events 
are  both  functions  of  time,  and  he  immediately  proceeds  to  form 
the  derivative  of  the  function  which  gives  the  subjective  phenomena 
as  depending  on  the  objective  phenomena;  to  this  (quantity  he 


*\\'.  P.  MoNTAGiE,  .4  Tkcury  oj  Time-l'erccption,  Amer.  Jo  of  Psychology. 
Vol.  XV,  1904,  pp.  1-13.  Mr.  Montague  came  very  near  to  considerations 
like  those  expounded  in  this  chapter,  but  instead  of  inquiring  into  the  principles 
of  the  problem  he  unfortunately  undertook  to  solve  a  very  special  and  very 
difficult  question.  Some  of  the  weaknesses  of  Mr.  Montague's  argumenta- 
tion are  shown  in  the  author's  note  on  "  The  Application  of  Calculus  to  Menial 
Phenomena,"  The  Jo  of  Phil.  Psych,  and  Scient.  Methods,  \o\.  II,  1905,  pp. 
16-18,  but  only  a  general  allusion  is  made  to  the  fact  that  the  weaknesses  of 
Mr.  Montague's  theory  are  the  necessary  outc(mie  of  a  certain  definite  hy- 
pothesis on  the  relation  of  the  pliysic  and  the  psycliic. 


164  proble:ms  of  psychophysics 

gives  a  certain  psychological  interpretation/''  Thus  the  suppo- 
sition crept  in  that  the  dependence  of  mental  on  physical  events 
is  given  by  differentiable  functions.  This  supposition  excludes 
at  once  a  very  wide  range  of  possibilities,  and  it  contains  the  defi- 
nite statement  that  the  type  of  order  of  the  events  of  the  class 

A,  B,  C,  ....  is  the  same  as  that  of  the  events  a,  b,  c,  The 

reason  why  this  rather  obvious  fact  could  be  overlooked  lies  in 
the  peculiar  formulation  which  the  author  gives  to  his  problem. 
He  introduces  both  types  of  events  as  functions  of  time  and  con- 
cludes that  the  events  A,  B,  C,  ....  must  be  functions  of  a,  b,  c, 

There  is  nothing  to  say  against  this  conclusion,  if  the  hpyothesis 
is  not  made  that  there  exists  a  derivative  of  this  function.  If 
one  makes  the  supposition  that  the  events  of  the  class  A,  B,  C, 
....  and  those  of  the  class  a,  b,  c,  ....  are  analytic  functions  of  time, 
one  must  necessarily  accept  the  conclusion  that  the  events  A,  B, 
C,  ....  are  analytic  functions  of  the  events  a,  b,  c, This  suppo- 
sition certainly  had  not  been  made,  if  it  had  been  seen  what  the. 
bearing  of  the  hypothesis  is  that  mental  events  are  analytic  func- 
tions of  physical  phenomena. 

Physics  leads  necessarily  to  analytic  functions  for  the  descrip- 
tion of  natural  phenomena,  because  they  are  the  solutions  of 
differential  equations  and  it  is  obvious  that  all  the  problems  which 
deal  with  events  which  cannot  be  described  by  them  must  escape 
notice.  This  caused  the  question  whether  the  description  of  phe- 
nomena as  supplied  by  physics  is  exhaustive,  and  on  what  ground 
the  supposition  rests  that  there  are  no  other  but  analytic  functions 
in  nature.  This  question  was  raised  by  F.  Klein, f  who  gave 
a  very  ingenious  answer  which  is  based  on  the  distinction 
between  functions  which  are  given  empirically  and  their  ideal 
mathematical  expression,  a  distinction  which  this  author  had 
introduced  previously. J     The  empirical  representation  of  a  func- 

*Mr.  Montague's  interpretation  of  the  derivative  did  riot  remain  uncontra- 
dicted; a  discussion  of  this  topic  may  be  found  in  E.  B.  Holt,  Jo  of  Phil. 
Psych,  and  Scient.  Methods,  Vol.  1,  1904,  pp.  320-323  and  Mr.  Montague's 
answer  in  the  same  journal,  Vol.  1,  1904,  pp.  378-382. 

tF.  Klein,  in  his  lectures  of  1901,  Vorlesungcn  ilber  Differential  und  In- 
iegralrechnung,  eine  Revision  der  Principien,  ed.  1907,  pp.  129-139. 

JF.  Klein,  Uber  den  allgemeinen  Functionsbegriff  und  desscn  Darsiellung 
durch  eine  willkurliche  Curve,  Math.  Ann.  Vol.  22,  1883,  pp.  249-259. 


THE   PSYCHOMETRIC  FUNCTIONS  165 

tion  is  never  an  ideal  curve  but  an  area  of  finite  extension  ami 
what  is  given  empirically  are  neither  analytic  nor  non-analytic 
functions,  but  only  approximations.  The  distinction  between 
analytic  and  non-analytic  functions  refers  merely  to  their  idealized 
mathematical  representation.  This  solution  cannot  be  discussed 
here  and  also  another  solution  given  by  J.  Boussinesq  can  be  but 
briefly  mentioned.*  Boussinesq  starts  from  the  view  that  the 
exact  description  of  a  phenomenonns  given  by  its  differential 
equation  with  which  its  integral  is  equivalent,  because  purely 
logical  processes  are  needed  for  finding  the  integral  of  a  differen- 
tial equation.  These  solutions  are  given  by  analytic  functions 
which  determine  the  process  everywhere  except  at  certain  points, 
the  branch  points,  where  the  choice  between  different  courses 
comes  in.  The  course  of  events  is  not  defined  at  these  points 
and  it  is  there,  at  the  branch  points,  that  Boussinesq  looks  out 
for  the  manifestations  of  mental  life,  more  especially  of  free  will, 
which  manifests  itself  by  the  undetermined  choice  between  equal 
possibilities  represented  by  the  branches  of  the  curve.  This 
view  has  not  found  many  followers  and  it  stands  out  more  as  a 
curious  example  of  the  ingenuity  of  its  promoter. 

The  third  attempt  at  a  solution  of  our  problem  has  a  direct 
bearing  on  the  question  treated  here,  and  this  view  is  formidable 
by  the  number  of  distinguished  followers  it  has  found.  The  starting 
point  for  these  considerations  is  found  in  the  fact  that  certain 
phenomena  which  are  the  product  of  human  will-decision  show 
a  remarkable  stability  in  their  averages,  if  they  are  taken  in  large 
groups,  no  matter  how  much  chance  may  have  influenced  the 
single  decisions.     These  results  of  demography  and  statistics  of 


*J.  BoussixESQ,  Co7iciliation  du  veritable  d Hcrmi ni  ime  mccanique  etc. 
Recucil  de  la  Societe  des  Sciences  dc  Lilies,  Vol.  VI,  187S  and  C.  R.  Vol.  XCIV, 
p.  208.  Boussinesq  treats  the  same  problem  in  his  book  "Application  des 
poteniicls  a  I' etude  de  I' equilibre  et  du  monvemcnt,"  1885,  pp.  699-704,  where 
he  attempts  to  show  that  functions  which  have  no  derivative  can  be  treated 
in  the  same  way  as  other  functions  by  considering  functions  which  differ  from 
them  only  in  a  very  slight  degree.  In  this  attempt  he  approaches  Klein's 
conception  of  a  "  Functiotissireifen"  by  making  it  a  matter  of  preference 
which  one  of  a  certain  group  of  functions,  which  difTer  from  a  given  function 
by  less  t'lan  a  certain  quantity,  one  will  treat. 


166  PROBLEMS  OF  PSYCHOPHYSICS 

morals  have  always  aroused  great  interest  and  were  frequently 
used  as  arguments  in  favor  of  universal  mechanical  causality 
against  free  will.  Instead  of  drawing  these  conclusions  one  must 
stop  considering  the  conditions  under  which  events,  which  are 
taken  in  large  groups,  can  possibly  show  regularity  in  the  mean 
values.  In  regard  to  this  question  there  exists  a  famous  propo- 
sition due  to  Tchebitcheff,  one  of  the  promoters  of  the  view  in 
question,  that  the  chief  condition  for  the  applicability  of  mean 
values  is  that  the  single  events  must  be  independent  from  each 
other.  The  results  admit  of  a  correct  interpretation  if  the  events 
are  independent  from  each  other,  but  if  they  are  not  independent 
i.  e.  if  they  are  causally  connected,  one  cannot  apply  the  theorems 
of  the  calculus  of  probabilities.  From  the  fact  that  we  find  regu- 
larity in  large  groups  of  human  actions  we  must  conclude  that 
they  are  independent  i.  e.  that  they  are  not  in  a  causal  relation, 
and  since  all  events  of  the  causal  chain  are  inter-connected  we 
must  conclude  that  will  decisions  are  not  causally  necessitated. 
The  law  of  causality  is,  therefore,  not  the  only  one  and  not  all 
events  of  nature  are  characterized  by  analytic  functions.  In  this 
way  the  old  argument  against  free  will  is  turned  into  an  argument 
for  it.  The  argumentation  as  expounded  is  due  to  the  so-called 
Moscow  school  of  idealism  as  represented  by  P.  L.  Tchebitcheff, 
Nekrassow,  W.  G.  Alexejeff,  and  the  head  of  the  school  N.  W. 
Bugajeff.  These  men  have  also  materially  promoted  the  study 
of  the  "half-analytic"  functions  which  originate  by  the  combin- 
ation of  analytic  with  non-analytic  functions,  and  in  which 
these  authors  see  a  more  adequate  expression  of  the  relation  of 
the  physic  and  the  psychic,  and  important  works  on  the  theory 
of  probabilities  are  due  to  them.  Alexejeff  has  the  merit  of  mak- 
ing these  investigation  generally  accessible  by  several  CJerman 
papers.* 

.*\V.  G.  Alexejeff,  Ubcr  die  Entwicklung  des  Begnffes  der  hoherenarith- 
mologischcn  Geselzmdssigk-eiten  in  Natur-und  Geistesivissenscliaft,  Vierteljahrs- 
schrift  f.  wiss.  Phil.  u.  Soz.,  Vol.  28,  1904,  pp.  72-93;  "N.  W.  Bugajew  und  die 
idealistischen  Probleme  der  Moskaiier  mathematischen  Scliule,"  the  same  jour- 
nal, Vol.  29,  1905,  pp.  335-367;  and  "Die  arithmologische  und  wakrschein- 
Uchkeitstheoretische  Causalitdt  etc.,"  Ztsclir.  f.  Philosotyhie  u.  Padagogik,  Vol. 
14,  (2.)  1906,  pp.  50-55. 


THE  PSYCHOMETRIC  FUNCTION'S  107 

This  argument  proves  too  much,  because  it  applies  to  all  kind 
of  random  events  among  which  there  are  many  classes  of  events 
which  doubtlessly  are  causally  necessitated.  There  are  events 
which  we  can  submit  successfully  to  the  treatment  by  the  calculus 
of  probabilities,  although  the  single  events  are  causally  necessitated 
or  even  logically  determined.  There  cannot  be  any  doubt  in  these 
cases  that  an  independence  of  the  single  events  does  not  exist  in 
any  absolute  sense  of  the  word,  but  large  groups  of  events  comply 
with  the  requirements  of  the  calculus  of  probabilities  in  a  very  high 
degree.  Let  us  take  the  following  examples.  Whether  high  water 
arrives  at  London  Bridge  in  the  first,  second,  third  or  fourth 
quarter  of  an  hour  of  the  morning  is  the  very  type  of  an  event 
which  is  causally  necessitated.  The  single  events  depend  entirely 
on  physical  conditions,  but  the  results  of  a  long  series  of  obser- 
vations conform,  nevertheless,  with  the  requirements  of  the  cal- 
culus in  a  verv  high  degree.*  The  following  example  of  events 
which  comply  almost  perfectly  with  the  requirements  of  the  cal- 
culus of  probabilities,  although  they  are  logically  determined, 
is  due  to  Bruns.  The  distribution  of  zeros  at  the  last  place  of 
the  logarithms  in  the  columns  (of  60  numbers  each)  of  Vega's 
Thesaurus  Logarithmorum  shows  that  the  numbers  of  those  col- 
umns in  which  zero  occurs  1,  2,  ....  60  times  is  very  nearly  such 
as  it  ought  to  be,  if  the  occurrence  of  the  numbers  at  the  last  place 
were  entirely  a  matter  of  chance. f     It  is  quite  obvious  that  the 


*F.  Y.  Edgevvorth,  The  Law  of  Error,  Part  II,  Transactions  of  the  Cam- 
bridge Philosophical  Society,  Vol.  20,  190-5,  p.  128,  129. 

fH.  Bruns,  Wahrscheinlichkeiisrechnuug  iind  Kollektivmasslclire,  1905, 
pp.  8,  279  sqs.  Bruns  had  chosen  this  example  on  purpose,  because  in  the 
case  of  the  occurrence  of  the  dilTerent  numerals  at  certain  places  of  the  h^g- 
arithms  one  cannot  possibly  speak  of  chance  in  the  common  sense  of  the  word. 
The  observation  that  tables  of  logarithms  are  material  for  the  study  of  chance 
events  was  made  by  Gauss,  Einige  Bemerkungen  zu  Vegas  Thesaurus  Log- 
arithmorum, ]]'erke,  Vol.  Ill,  p.  260.  Gauss  remarks  that  the  last  decimal 
of  the  logarithms  of  the  tangent  cannot  be  equal  to  the  last  decimal  of  the 
difference  of  the  logarithms  of  the  sine  and  cosine,  if  the  deviations  of  the  loga- 
rithms of  the  sine  and  of  the  cosine  from  the  correct  values  have  not  the  same 
sign,  and  if  their  sum  is  greater  than  \.  The  probability  of  this  event  is  \. 
Gauss  found  by  counting  over  the  tables  of  K(')hler  that  this  event  happened 
in  2.50  out  of  900  cases. 


168  PROBLEMS  OF  PSYCHOPHYSICS 

occurrence  of  a  certain  numeral  at  the  tenth  phice  of  the  logarithms 
of  consecutive  numbers  is  not  an  event  which  has  random  char- 
acter in  an  absolute  sense  of  the  word,  since  we  can  determine 
the  event  in  every  case  by  the  rules  of  arithmetic.  The  fact  is 
that  randomness  of  the  events,  which  can  be  submitted  to  the 
treatment  by  the  calculus  of  probabilities,  consists  in  the  absence 
of  a  knowable  law.  This,  as  a  matter  of  fact,  is  the  conclusion 
from  the  test  whether  a  certain  group  of  events  has  random 
character  or  whether  it  has  not,  that  the  agreement  with  the  rules 
of  the  calculus  of  probabilities  indicates  the  absence  of  a  recog- 
nizable law,  and  the  lack  of  agreement  with  the  results  of  the 
calculus  indicates  the  presence  of  a  constant  influence  which  may 
be  investigated.  This  type  of  probability,  w'hich  may  be  called 
randomness  by  excessive  complication,  has  been  recognized  al- 
ready by  Kepler,  as  it  was  pointed  out  recently  by  Bruns.*  It 
is  exactly  this  type  of  probability  which  we  use  for  the  working 
out  of  the  results  of  our  experiments  on  lifted  weights.  Every 
single  event  is  causally  necessitated,  but  its  conditions  are  so  com- 
plicated that  we  must  be  satisfied  with  this  very  general  type  of 
relations  which  form  the  propositions  of  the  calculus  of  probabil- 
ities. A  mathematical  probability  is  defined  as  a  fraction  the 
numerator  of  which  gives  the  number  of  cases  which  are  favorable 
to  the  event,  and  the  denominator  of  which  gives  the  total  num- 
ber or  possible  cases.  The  outcome  of  every  single  case  must  be 
well  defined.  We  have  seen  above  that  the  numbers  of  relative 
frequency  for  the  judgments  of  different  types  on  the  comparison 
of  two  weights  have  the  formal  and  material  character  of  prob- 

*H.  Bruns,  /.  c.  p.  7.  The  place  in  question  is  J.  Kepler,  Dc  stclla 
nova  in  pede  Serpentarii,  Opera,  ed.,  Frisch,  Vol.  II,  p.  714  "Improvidi  sunt, 
qui  hos  (tesserarum  jactus)  plane  fortuitos,  hoc  est  avairiovr^  esse  putant:  sin 
autem  suum  casum  omni  causa  privant,  nondum  ejus  exemplum  dixerunt 
in  lessens.  Quare  hoc  jactu  Venus  cecidit,  illo  canis?  Nimirum  lusor  hac 
vice  tessellam  alio  latere  aripuit,  aliter  manu  condidit,  aliter  intus  agitavit, 
alio  impetu  animi  manusve  projecit,  aliter  interflavit  aura,  alio  loco  alvei 
impegit.  Nihil  hie  est,  quod  sua  causa  caruerit,  siquis  ista  subtilia  possitcon- 
sectari. "  The  example  of  the  acetarium  on  the  previous  pages  of  Kepler 
serves  as  illustration  against  certain  ancient  systems  of  philosophy  that  all 
possible  combinations  of  events  must  be  exhausted  in  an  infinite  interval  of 
time. 


THE    PSYCHOMETRIC   FUNCTIONS  169 

abilities.  The  iiuinber  of  cases  which  are  favorable  to  the  event, 
e.  g.  to  the  formation  of  the  judgment  "heavier",  is  represented 
by  a  finite  or  infinite  number  of  groups  of  conditions  each  one  of 
which  leads  necessarily  to  the  formation  of  the  judgment  that  the 
second  weight  is  heavier.  The  number  of  possible  cases  is  rep- 
resented by  the  totality  of  conditions  in  which  a  judgment  is 
given  according  to  the  rules  of  the  experiments.  The  ratio  of 
these  two  numbers  gives  the  probability  in  question.  Only  under 
the  supposition  that  there  exists  a  definite  (finite  or  infinite)  num- 
ber of  possibilities  each  one  of  which  leads  necessarily  to  a  certain 
result  can  we  form  the  notion  of  the  probability  of  a  "heavier"- 
judgment.  We  conclude  from  the  fact  that  the  notion  of  a  math- 
ematical probability  can  be  successfully  applied  to  physical 
events,  that  they  have  very  nearly  the  character  of  those  events 
which  we  treat  in  the  system  of  idealized  propositions  which 
we  call  the  calculus  of  probabilities.  The  proposition  that  a 
certain  well  defined  group  of  physical  events  has  the  character  of 
random  events  in  the  mathematical  sense  of  the  word  warrants 
the  application  of  the  rules  of  the  calculus  to  these  events.  This 
statement  plays  the  same  role  for  this  group  of  events  as  the  state- 
ment that  physical  space  is  a  three-dimensional  Euclidean  mani- 
foldness  plays  for  the  application  of  ordinary  geometry  to  physi- 
cal space.  The  conformity  of  events  with  the  rules  of  the  calcu- 
lus of  probabilities  is  so  far  from  being  an  argument  for  the  lack 
of  causation  of  these  events,  that  it  is  an  argument  for  uniform 
causality.  No  statement  about  the  future  outcome  of  obser- 
vations were  possible,  if  every  single  event  were  not  causally 
necessitated.* 

*This  conception  is  a  consequence  of  basing  the  idea  of  probability  on  the 
hypothetical  judgment,  as  it  was  done  by  Sigwart,  Logik,  Vol.  II,  p.  30.5- 
320.  For  the  logical  details  the  reader  must  be  referred  to  this  book  and  to 
the  treatises  of  K.  Stumpf,  Uber  den  Begriff  der  mathematischen  Wahrschein- 
lichkeit,  Bcr.  d.  bayr.  Ak.  {Phil.  Kl.)  1892;  J.  v.  Kries,  Die  Principien  der 
Wahrscheinlicbkeitsrechnung,  18S6;  H.  Bruns,  Wahrscheinlichkcitsrechnung 
und  Kollektivmasslehre,  1906,  and  to  the  repeatedly  quoted  works  of  E.  Czit- 
ber.  a  different  view  was  expressed  lately  by  Mr.  Gomperz,  Uber  dir 
W ahrscheinlichkeit  der  W illensentscheidungen,  Sitzungsberichie  d.  Kaiserlichen 
Akademie  der  Wissensckafien  zii  Wicn,  Phil.  Hist.  Kl.  Vol.  149,  III.  Abh.  who 
tried  to  find  a  new  conception  of  free  will  and  began  by  showing  that  the  sta- 


170  PROBLEMS  OF  PSYCHOPHYSICS 

The  process  of  finding  the  numerical  values  of  these  probabili- 
ties is  perfectly  well  defined  and  stands  on  the  same  level  with 
other  empirical  ol)servations.  The  theorem  of  Bernoulli  gives  the 
limits  of  exactitude  of  such  observations,  and  it  depends  on  our 
will  to  make  our  results  as  exact  as  we  choose.  The  empirical 
datum  of  a  determination  of  a  probability  is  a  realm  inside  of 
which  we  may  expect  the  result  with  a  given  probability.  The 
limits  of  the  exactitude  of  the  observation  can  be  made  smaller 
than  any  given  quantity  by  increasing  the  number  of  observa- 
tions. These  observations  may  be  made  for  different  intensities 
of  the  comparison  stimulus.  The  res,ult  of  these  experiments 
is  that  a  higher  probability  of  a  "greater "-judgments  corresponds 
to  higher  intensities  of  the  comparison  stimulus,  that  greater 
probabilities  of  "smaller "-judgments  correspond  to  smaller  in- 
tensities of  the  comparison  stimulus,  and  that  the  probabilities 
of  the  "equality "-judgments  increase  at  first  and  decrease  after 
having  attained  a  certain  maximum.  These  empirical  data  do 
not  differ  essentially  from  the  results  of  any  other  experimental 
investigation,  which  show  how  certain  quantities  depend  on  the 
variation  of  another  quantity.  These  data  are  subjected  to  a 
process  of  idealization,  the  first  step  of  which  is  to  assume  that 
an  absolutely  exact  observation  would  lead  to  a  definite  numerical 
value    and   an  hyjjothesis  is  made  which  these  values    are.     In 

tistical  regularities  do  not  prove  anything  in  favor  of  causal  necessitation  of 
the  single  events.  "  If  one  were  to  cast  a  die  60,000  times  every  year  and  if 
there  were  every  3'ear  among  the  results  approximately  10,000  aces,  nobody 
will  conclude  that  in  every  single  case  the  ace  had  to  come  out  necessarily." 
This  view,  which  Mr.  Gomperz  does  not  support  by  arguments,  is  refuted  by 
the  examples  mentioned  above,  where  events  about  the  causation  of  which 
there  cannot  be  the  slightest  doubt  comply  with  the  requirements  of  the  cal- 
culus of  probabilities.  Mr.  Gomperz's  papers  is  marred  by  some  slips  ("  an  al- 
most infinite  probability"  1.  c.  p.  13)  which  indicate  that  the  author  is  not  in- 
timately acquainted  with  the  theory  of  probabiHties,  a  supposition  which  is  sup- 
ported by  the  fact  that  a  "  mathematical  friend"  had  to  give  the  formula  p.l  6 
of  his  paper,   which  is  easily  found  by  calculating  the  point  of  intersection  of 

X  2  y  2  x-a  2  V 

^   '-(^)=land(— -)  +  (^-- 

s,  ^  as'  ■  s 


X  -=           y  2                        x-a  ^  y  ^  s , 

the  two  ecHpses  (  — )  +  ( )  =1  and  ( )-[-(_—)  =1  and  putting  —  =q 


Compare  the  review  of  Mr.  Gomperz's  paper  in  Ztchr.  f.  Psychologie,  Nov  1900 
Vol.  43,  p.  318. 


THE  PSYCHOMETllIC  FUNCTIONS  171 

the  case  of  observations  on  probabilities  the  result  is  determined 
by  the  theorem  of  Bernoulli,  in  empirical  sciences  this  result  is 
given  as  a  rule  by  the  most  probable  value  as  determined  by  the 
method  of  least  squares.  The  next  step  of  idealization  refers  to 
the  rule  of  dependence  between  the  variations  of  the  quantities 
observed.  The  observations  on  a  " greater"-]  udgment  show  that 
these  probabilities  increase  with  the  intensity  of  the  comparison 
stimuli.  This  observation  holds  only  for  the  finite  set  of  obser- 
vations which  were  really  made.  One  interpolates  between  the 
elements  of  this  finite  set  of  observations  the  elements  of  an  in- 
finite set  which  is  of  the  order  of  the  continuum.  The  elements 
of  the  infinite  set  are  supposed  to  follow  the  same  law  as  the  ob- 
served elements.  This  supposition  is  based  on  the  methodolog- 
ical principle  mentioned  above  that  unknown  phenomena  are  sup- 
posed to  follow  the  same  law  as  observed  phenomena  of  the  same 
class,  as  long  as  no  contrary  instances  are  known.  There  is  no 
difference  between  our  treatment  of  psychological  observations 
anil  the  methods  by  which  physical  observations  are  treated,  and 
we  may  say  in  general  that  the  mathematical  representation  of 
empirical  observations  is  nothing  else  than  an  idealization  of  ex- 
perience. 


APPENDIX 


173 


TABLE   1. 
Variations  of  the  Weights  used  i>f  the    experiments  (in  milligrams) 


^Corrected. 


u 

1 

2 

Weight 
(in  grams) 

February 
23,  07. 

March 
18,  07. 

Is" 

April 
8,07. 

<  2 

'1  ^ 
<?1 

o 
to 
>. 

a 

0    o 
> 

1 

100 

-  5 

-  5 

-  5 

-  9 

-  6 

-  4 

-  1 

-  8 

19 

2 

100 

+  2 

+    2 

+  2 

+  2 

+  3 

+  4 

+   5 

+   5 

3 

3 

100 

-3 

+   3 

+  5 

+  2 

-  7 

-  4 

-  6 

-  2 

31 

4 

100 







0 

0 

-  1 

0 

-  1 

4 

5 

100 

0 

-  4 

-  4 

-  3 

-  4 

-  3 

-  1 

-  3 

11 

10 

100 

-  4 

-  3 

-  6 

—  7 

-  8 

_  2 

-  4 

-  6 

16 

11 

100 

-  4 

-  4 

-  7 

-  3 

1^ 

-  3 

-  4 

-  7 

IS 

12 

100 

-  4 

+    4 

-  5 

-  5 

-  o 

-  6 

-  4 

-  9 

25 

6 

104 

^ 

-  4 

-  6 

-  6 

-  S 

—  7 

-  6 

-  8 

8 

7 

108 



-  4 

-  6 

-  6 

-  9 

-  9 

-  8 

-10 

8 

20 

92 



-  3 

-  3 

-  3 

-  7 

+  1 

-  7 

-  1 

26 

21 

96 



-  3 

-  3 

-  2 

-  5 

-  3 

-  9 

-  5 

16 

51 

84 



+  2 

+   3 

+  3 

+  2 

+  5 

-  3 

+   5 

21 

.52 

88 



-  4 

-  4 

-  4 

-  5 

-  1 

-10 

-  5 

19 

23 

90 



+  13* 

+    1 

+  1 

0 

-  2 

-  6 

-  3 

11 

24 

94 



-  4 

+  14* 

+  6 

+  6 

+  4 

-  4 

+   2 

40 

25 

98 



+  14* 

-  2 

+  2 

-  1 

-  2 

+    2 

0 

16 

26 

102 



+   3 

+   4 

+  5 

+  5 

0 

+   4 

+   5 

12 

TABLE  2. 


Date  of  observation 

I 

II 

III 

IV 

April         26.           07 

100.308 

104.364 

103.003 

114-227 

29, 

100.423 

104.488 

103.434 

114.781 

May           2. 

100.299 

104.360 

103.159 

114.709 

6. 

100.357 

104.439 

103.357 

114.747 

■    "            10, 

100.654 

104.704 

104.280 

115.342 

Observed  variations 


Sum  of  variations 


1.259 


174 


PROBLEMS  OF   PSYCHOPHYSICS 


M     in 

m 


- 

-' 

O 

o 

o 

o  c 

■^ 

-^ 

° 

CO 

^ 

Tf 
^ 

o 

CO 

_M 

o 

'-' 

O 

■-^ 

o  w 

o 

'-' 

o 

^ 

"5 

•^ 

CO 

(N 

IM    (N 

-^ 

o 

o 

2 

H 

O 

lO 

CO 

CO 

(N    ■«< 

'^ 

-^ 

o 

s 

„       I       OrHO-^Or-i.-lC^JC 


(M    1--   I^   CO   ■* 


O-HiofNtNOOO      I     .-I 


I~(NTq(NTj(cOU5IMCO 


OOC^lCOt^tOlOiiilMCO     I     —I 


o 
o 

-  1 

CO  M 

X 

r^  r^  CO  »-<  lO  CD    1 

lO 

>o 

-I 

CO  to 

X 

_|     -H     rt    r-^    OS    CD      1 
(M    ^    <M    CO    O-l    C^ 

CD 

X 

^ 

CO   o 

>o 

■*    O    t^    rt    CO    •fl^ 

CO 

00  oi 

X    (N    CD    1^    O    ■* 
--c    (M    rt           rt    rt 

CO 

CD 

H 

CO    C^l 

CO 

!M    C^l    CO    X    CO    X 

01  CO  C-)        — 1  rt 

C5 

c 

CO 
05 

" 

o   t^ 

o 

M    — ■    CO    CD    CO    LO 

■-H     IM     <N     ,-1     r-<     C>1 

c 

■^ 

X    (35 

w 

05    CO    CO    •*    ■*    lO 

M 

O    X 

X 

O    O    -M    O    --I    — 1 

c 
c 

M 
^ 

H 

CD    CO 

co"-* 

(N    0-1 

lO 

O    t-    3!    O    O    O 

CO 

CO 

05    CO    -H    o    O    O 
rt    r-l    (N    (M    <N    C^l 

35 

" 

CD    CO 
(N    CO 

10 

ic  (M  t^  lo  in  CO 

rH    N    (N    CO    CO    ■* 

CD 
(M 

XI 

■*    1^ 

'^ 

CD    CD    lO    CO    X    O 

O 

W) 

^X 

X 

^    CO    "—I    OS   CO    CO 

CO 
OS 

J3 

C!    CI 

CO 

•O    CO    t^   CO    ■*    ■* 

CO 

lO 

H 

p  o 

-* 
M 

33    C-l    X    IM    t^   r^ 

C)      C^)      — 1     r-t 

00 
00 

- 

CO    ■* 

CO 

O    t^    CO    -^    C    X 
^    ^    tJ<    ^    ^    ^ 

X 

CO 

o 

^ 

-o 

o 

CO   ^   O   ^   CO   o 

_M 

LO    lO 

■* 

■>}<    N    CO    •*    "O    W 

60 

J3 

(N    O 

CO 

CO    O    ^    rt    (N    O 

CI 

H 

t^    lO 

t^ 

t^    IM    •*    lO    t^    N 

CD 

1< 

00 

- 

CO    t^ 

CD 

CO  X  >o  X  r^  X 

0) 

o  o 

o 

o   o   O   C   C   -H 

^ 

CO 

■*  M  CT  e<i  CO  -H 

!M 

BO 

o,^ 

'-' 

O  O  CO  o  o  o 

I- 

X 

H 

•*  CO 

T(< 

m  cj  lo  c^)  CO  --I 

c 

L) 
-. 

(M 

> 

^    (%]    ^    N    -1   N 

^ 

2 


- 

° 

!M 

o  o 

(N 

CO 

c  o 

o 

1    '^ 

.« 

r- 

■* 

C-l 

CD 

lO    CO 

Til     T)( 

■* 

1    — 

1    5< 

_bfl 

IM 

-' 

CO 

-' 

CO    O) 

IM 

1  s 

BO 

-^ 

lO 

■*    (M 

CO 

O 

IM    N 

^ 

1  s 

H 

CO 

CO 

lO   CO 

CO 

-^ 

lO  ■* 

CO 

1 5§ 

^   I  c-l  t^  CO  0>  CD  !-•  O  O  (M 


"bo~ 


— I  c-i  o  ci  ^^  rt  rt  CO  -a< 


CO  lO  C^  IM  t>  lO 


1 

rH 

,^ 

O) 

CO 

IM 

o 

05 

o 

,_) 

1    •* 

'~   \ 

'^ 

'^ 

"^ 

^ 

c-l 

'"' 

o 

1 

(M 

IM 

OS 

Tti 

lO 

,_! 

w 

■* 

-* 

1   <» 

X!     1 

1 

(M 

C4 

(M 

IM 

CI 

CI 

CO 

CO 

CO 
1      CI 

BO     1 

-^ 

CD 

-* 

t~ 

lO 

CO 

t~ 

CO 

■* 

1  % 

M     1 

CO 

rt 

lO 

O 

X 

CO 

1^ 

IP 

^ 

1   '^ 

J=      1 
,        1 

•-H 

1     CD 

1- 

o 

,-^ 

CO 

-rs 

-^ 

^ 

o 

1    °^ 

^      1 

'^ 

'"' 

""^ 

1  2 

•-icoior-t^iocO'-it^ 


TtiCOOCOCO'OXXiO     I     -^ 


CDOCO-^t^wSt-COCO 


I  O  CO  CI  o 

H      ,-1  r-l  rt  -H 


•^  ic  CO  CO  O   I  "Oi 

CO  CO  CO  CO  CO    —I 


^  \ 

Tj<    CO    CI 

2  °^ 

CO  ■* 

CO  c    1    c 

i?  1 

CO  CO  CI 

Cl   lO 

in  ■* 

t-  CI    1    a 

1    c» 

1  \ 

CD  in  lo 

in  CO 

t^   CD 

t-x    1    c 

C5Xr-t^xcio^o 


^  -H  -    I   X 


I   OCO'^lO'-lCO^^Cl'^ 


I  i-lOOO'-l--iC0'-iC]   I  --< 


d'^cocoinoco'^ci 


ci  CO 


CO  O 
CO  CO 


CO  ■*  CO  CO  Cl 


X  CO  CO  t^  ■^  I  '^ 


I      CDt^O5C0O3XC0lOt>- 


-H    d    O    O    --<    --I    "-I 


1-1  CO  ^ 

-^oocO'-''-icicici    I    cT 


ClCOClO'-i— <cio 


CO   C    O    lO   — '    CI    CO   Tj*    M 


I       Cli      ClJ 

1    cq  — 


rt    „    ;.)    rf    CI    ^ 
CI 


^ 


APPENDIX 


175 


^  1 

o 

O    O   CO 

o 

i-l    O   <N 

o 

1  '^ 

H    1 

-• 

O   O    CO 

o 

rt   o   <N 

o 

"- 

-    1 

-^ 

rt     T)<     CO 

lO 

r-H     C-l     lO 

lO 

s 

-1 

■*     Tj<     ^ 

■«<  Tjl  ■* 

■* 

03  a;  •* 

o 

^   1 
>   1 

'-' 

O     O     -H 

o 

o  o  o 

o 

1  ^ 

(M 

O    (M    O 

•* 

O    O    -H 

o 

1  2 

C0"3WO'*OO'-iO     I     -H 


M    Ol    Tj(    w   O 
— I    — I    (>)   M   CO 


■*OOt^OC0OOC 


CO   "O   O   00   5D   Ol   C 


rj  uo  CO  M  C5 


O   — '    'M    C5   O      I     O 
TT    '«<    •*    CO    ■*  —I 


X 

i^  r~  t^  -^  o  -H  t^  o  o 

o 

o 

s 

-<OOCTOO"00 

^ 

60 

OOOMOOOO-^O 

^ 

,-(OO'<l'O00— ir-iO 


X  r^  00    I    -H 


ri 

C-l    S3    rt    C)    T-i    (M    W    M    ^ 

Si 

to 

CO'^OCOOOOOO 

^ 

60 
J3 

00C3OO^OO-i-l 

g 

■-iCOCO— lOO'-l'-i 


„ 

O 

o 

o 

o 

o 

iO 

o 

CO 

if 

CO 

lO 

o 

IM 

o 

o 

o 

^ 

o 

;:; 

00 

'-' 

05 

o 

'-' 

o 

c 

o 

o 

° 

^ 

^ 

'-' 

'- 

o 

CO 

o 

o 

= 

o 

o 

lO 

H 

^, 

CO 

o 

■* 

o 

o 

o 

o 

o 

03 

- 

o 
to 

X 

■* 

LO 

o 
>-o 

o 

s 

s 

o 

o 
to 

00 
"J" 
•"J" 

60 

= 

o 

o 

o 

o 

o 

o 

o 

o 

° 

■<i' 
00 

° 

o 

o 

o 

o 

o 

o 

o 

o 

° 

^ 

° 

(N 

o 

o 

o 

o 

o 

o 

o 
"o" 

(N 

C-i 

° 

(M 

o 

o 

o 

o 

o 

o 

N 

I     rt    ■«  i?  -^  <^'  "^  '^'  "^  <M 


:^ 


o 

-  1 

N  CO  -H  o  o  o    1 

CO 

■^  1 

t^  CO  d  lO  ci  03 

03 
Ol 

M     1 

-H  rH  o  -^  o  o    1 

CO 

^ : 

O   CO   CO    ■*    -H    1-1     1 

Ol 

~lO~ 

~i.O 

^  1 

-H     ■*     CO     lO     -H     -H       1 

o 

-  1 

■*    IM    -1    CO    rt    ■*      1 

X 

lO  O  O   C<l   O   ^1     1 

CD 
Ol 

5 

rt   O  rt  CO  o  o     1 

lO 

O   CO    (M    IM   CO   U3     1 

lO 

e 

-H    CO   CO    1^5    CO   "3     1 

o 

Ol 

o 

o 

- 

rt    TS    M    t^    T*    lO     1 

CO 

o 

XI 

03  c;  C3  t^  X  —    1 

tM    C>1    01    CO    .-o    0) 

CO 

X 

_ao 

00  CO  CO  -^  «  o     1 

CO 
Ol 

IM    •*    CO   !N    CD   OO     1 

Ol 

^    1 

O   t^   C3   CO    00   •<J<     1 

Ol 

CO 

CO 
■03 

CO   "3   O   O    03   CO     1 
rt    IM    r-i    C-I    '-I    CO 

Xi 

"5   •*    I^    0}    -"l"    OO     1 
Cj    rt    rt    rt    (M             , 

o 

£S 

•<J<    I^   C3    ■*    CO    CO     1 

CO 

^5 

^ 
^ 

lO  ■*!  in  t^  •*  CO    1 

C3    -H    ■*    •"!    t^    03      1 

X 
Ol 

o 

03 

X 
00 

00 

lo  ■*  o  o  -1  m    1 

CO   ■»  CO  CO   •*   •* 

o 
CO 

IM 

■^ 

t^  -^  M  »  in  1-1    1 

to 
CO 

^ 

CO    Ol    •^    ■^   CO   CO     1 

Ol 
Ol 

s 

IM  CO   CO  CO   -^  ^     1 

w 

^ 

„ 

00  >o  t-  t-  ■*  •*    1 

0)    00    00    01    00   C3     1 

Tjl     TH     Tf     Tl<     ■*!     Tt< 

lO 

_co_ 

Ol 

■^ 

O)   O  O  CO   Ol   1-1     1 

X 

i? 

■<J<  O  O  Ol   o  o     1 

CO 

^ 

H 

(M    (N    IM    CO    O   O     1 

03 

to    01    M    "5    O    O     1 

o 

O  C5  t^   00  ■*   c 

X 
X 
01 

J3 

O  O  Ol   rt  ■*   c 

o c 

60 

O  O  O  O  >-i  c 

H 

o  ^  -  ^  (N  e 

c 

Z 

,H  iM  -■  IN  '^  e« 

-'  «  ^  «  >  > 

176 


PROBLEMS  OF  PSYCHOPHYSICS 


o 
o 

- 

o 

-^ 

^ 

CI 

- 

-^ 

1  to 

Si 

5< 

00 

X 

CI 

03 

CO 

00 

CI 

_5f 

'^ 

o 

o 

c 

o 

o 

'^ 

43 

00 

^ 

^ 

o 

o 

o 

1  s 

H 

a> 

-^ 

'^ 

o 

o 

o 

1  s 

" 

■* 

IN 

(N 

'^ 

'^ 

o 

1  2 

1 « 

CI 

,« 

§ 

■<1< 

to 

00 

_ho 

o 

o 

'-' 

o 

o 

o 

-- 

t- 

'" 

CI 

^ 

o 

t^ 

IM 

cq 

'^ 

1 

o 

o 

^ 

t^ 

^ 

::! 

o 

CI 

CO 

'-* 

CO 

>f5 

^ 

to 

o 

CO 

05 
C) 

M 

■* 

« 
■* 

•* 

§ 

_M 

'^ 

■* 

o 

o 

'^ 

'^ 

^ 

^ 

to 

in 

o 

'-' 

■* 

o 

^ 

^ 

t- 

03 

2 

o 

0\ 

C] 

CO 

o 

^ 
fe 

to 

05 

- 

o 

^ 

X 

to 

w 

CO 

CI 

CO 

CO 

2 

W) 

o 

•* 

'^ 

o 

CI 

CO 

2 

^ 

cc 

"^ 

to 

o 

to 

CO 

05 
LO 

^ 

CO 

•o 

r- 

o 

00 

to 

(M 

'OS 

00 
00 

o 

CO 

I- 

CO 

■* 

05 

CI 

1  ^ 

1  c3 

•« 

00 

to 

o 

00 

•* 

CO 

s 

CO 

CO 

'^ 

o 

OS 

CO 

2 

CO 

to 

CO 

'-' 

00 

CI 

g5 

^^ 

to 

~to~ 

"oo' 

o 

^ 

C) 

•^ 

o 

'-" 

= 

CI 

CO 

'-' 

'^ 

^ 

^ 

o 

o 

o 

N 

'-' 

^ 

^ 

CO 

I^ 

o 

o 

o 

'^ 

^ 

^ 

•* 

t^ 

o 

o 

C) 

CI 

o 

00 

H 

00 

■<1< 

o 

•o 

05 

to 

to 

00 
GO 
CI 

•«  1 

CI 

o 

o 

^ 

'-' 

■* 

00 

^  1 

o 

o 

^ 

o 

CI 

o 

CO 

s 

o 

o 

o 

o 

-^ 

o 

'-' 

^  1 

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APPENDIX 


177 


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PROBLEMS  OF  PSYCHOPHYSICS 


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1  ^ 

APPENDIX 

TABLE   17. 


IS! 


P.E. 


0 

50 

0.01923 

0.9S077 

0.00000 

1 

49 

.03846 

.96154 

.01335 

2 

4S 

.05769 

.94231 

.01869 

3 

47 

.07692 

.92308 

.02265 

4 

46 

.09615 

.90385 

.02588 

5 

45 

.11538 

.88462 

.02862 

6 

44 

.13462 

.86538 

.03100 

7 

43 

.15385 

.84615 

.03310 

8 

42 

.17308 

.82692 

.03497 

9 

41 

.19231 

.80769 

.03665 

10 

40 

.21154 

.78846 

.03816 

11 

39 

.23077 

.76923 

.03951 

12 

38 

.25000 

.75000 

.04074 

13 

37 

.26923 

.73077 

.04184 

14 

36 

.28846 

.71154 

.04283 

15 

35 

.30769 

.69231 

.04371 

16 

34 

.32692 

.67308 

.04450 

17 

33 

.34614 

.6.5386 

.04519 

18 

32 

.36538 

.63462 

.04579 

19 

31 

.38462 

.61538 

.04630 

20 

30 

.40385 

.59615 

.04673 

21 

29 

.42308 

.57692 

.04708 

22 

28 

.44231 

.55769 

.04735 

23 

27 

.46154 

.53846 

.04754 

24 

26 

.48077 

.51923 

.04766 

25 

25 

.50000 

.50000 

.04769 

TABLE  18. 
Coefficients  of  Divergence. 


SUBJECT  I. 


h' 

h 

Q 

84  h 

106.72 

106.06 

1.00 

84  1 

20.37 

32.35 

0.63 

88  h 

35.71 

28.87 

1.24 

88  1 

14.23 

7.19 

1.98 

92  h 

17.57 

13.13 

1.34 

92  1 

10.13 

4.23 

2.39 

96  h 

12.03 

5.77 

2.08 

96  1 

10.59 

5.99 

1.77 

100  h 

10.15 

4.28 

2.37 

100  1 

15.27 

13.25 

1.15 

104  h 

16.35 

16.01 

1.02 

104  1 

43.65 

50.00 

0.87 

108  h 

21.05 

19.61 

1.07 

108  1 

61.29 

89.44 

0.68 

182 


PROBLEMS  OF  PSYCHOPHY8ICS 


TABLE  19. 
Coefficients  of  Divergence.     SUBJECT  IL 


h' 

h 

Q 

84  h 

33.94 

40.00 

0.85 

84  1 

20.02 

18.26 

1.10 

88  h 

32.41 

36.51 

0.89 

SS  1 

14.51 

23.10 

0.63 

92  h 

15.91 

11.45 

1.39 

92  1 

10.91 

11.47 

0.95 

96  h 

10.98 

14.43 

0.76 

96  I 

10.05 

11.16 

0.90 

100  h 

10.01 

7.10 

1.41 

100  1 

11.86 

8.98 

1.32 

104  h 

12.89 

6.14 

2.14 

104  1 

11.86 

8.29 

1.43 

108  h 

17.00 

17.82 

0.95 

108  I 

40.35 

29.49 

1.37 

TABLE  20. 

COEFFICIEXTS    OF    DIVERGENCE.        SUBJECT    III. 


84  h 

84  1 

75.54 

53.45 

1.41 

88  h 

32.41 

19.80 

1.64 

88  1 

24.27 

11.23 

2.16 

92  h 

19.45 

8.75 

2.22 

92  1 

14.81 

4.26 

3.48 

96  h 

12.03 

4.84 

2.48 

96  1 

10.91 

3.35 

3.26 

100  h 

10.02 

5.39 

1.86 

100  1 

10.30 

6.36 

1.62 

104  h 

16.20 

12.91 

1.25 

104  1 

21.05 

19.61 

1.07 

108  h 

24.87 

12.35 

2.01 

-08  1 

31.02 

19.61 

1.58  . 

APPENDIX 


183 


TABLE  21. 
Coefficients  of  Divergence.     SUBJECT  IV. 


84 

h 

S4 

88 

h 

88 

92 

h 

92 

96 

96 

100 

100 

104 

104 

108 

108 

h' 

h 

Q 

33.14 

22.14 

1.50 

25.51 

15.50 

1.65 

31.02 

29.36 

1.06 

18.79 

10.89 

1.72 

15.57 

8.73 

1.78 

11.82 

5.98 

1.98 

10.44 

6.12 

1.71 

10.07 

5.77 

1.75 

10.25 

5.01 

2.05 

12.28 

8.36 

1.47 

15.57 

16.57 

0.94 

22.94 

25.65 

0.89 

19.60 

15.08 

1.30 

35.72 

27.95 

1.28 

TABLE  22. 
Coefficients  of  Divergexce.     SUBJECT  V. 


h' 


84 

h 

84 

88 

h 

88 

92 

h 

92 

96 

h 

96 

100 

100 

104 

104 

108 

108 

31.02 

25.51 

33.15 

19.18 

13.64 

10.91 

10.00 

10.45 

10.72 

13.11 

14.87 

27.87 

21.62 

35.71 


23.57 

21.13 

30.43 

12.95 

9.73 

4.25 

3.78 

4.04 

3.31 

5.18 

4.58 

25.65 

18.70 

79.05 


1.31 

1.20 

1.09 

1.48 

1.40 

2.56 

2.64 

2.58 

3.23 

2.53 

3.24 

1.08 

1.15 

0.45 


184 


PROBLEMS  OF  PSYCHOPHYSICS 


TABLE  23. 
Coefficients  of  Divergence.     SUBJECT  VI. 


h' 

h 

Q 

84  h 

61.29 

43.85 

1.39 

84  1 

26.45 

12.62 

2.09 

88  h 

33.14 

14.74 

2.25 

88  1 

14.55 

11.56 

1.26 

92  h 

20.53 

12.70 

1.62 

92  1 

11.10 

7.14 

1.56 

96  h 

12.83 

6.22 

2.06 

96  1 

10.00 

9.35 

1.07 

100  h 

10.13 

6.83 

1.48 

100  1 

12.35 

15.14 

0.81 

104  h 

11.44 

5.43 

2.09 

104  1 

20.04 

14.14 

1.42 

108  h 

15.57 

9.02 

1.72 

108  1 

35.72 

32.27 

1.10 

TABLE  24. 
Coefficients  of  Divergence.     SUBJECT  VII. 


h' 

h 

0 

84  h 

61.29 

111.8 

0.55 

84  1 

35.71 

79.06 

0.45 

88  h 

61.29 

70.71 

0.87 

88  1 

16.43 

24.69 

0.66 

92  h 

20.04 

13.71 

1.46 

92  1 

11.40 

6.78 

1.6S 

96  h 

13.29 

7.70 

1.73 

96  1 

10.04 

8.69 

1.15 

100  h 

10.08 

6.52 

1.54 

100  1 

11.31 

6.76 

1.67 

104  h 

11.65 

7.51 

1.55 

104  1 

21.05 

1.83 

1.64 

108  h 

13.01 

6.78 

1.92 

108  1 

25.52 

21.13 

1.20 

APPENDIX 


185 


TABLE  25. 
Relative  Frequency  of  "Heavier"  Judgments  in  the  Different  Columns. 


Subject 

1 

2 

3 

4 

5 

I 

0.5619 

0.5488 

0.5250 

0.5464 

0.5048 

II 

0.4824 

0.5004 

0.5333 

0.4929 

0.5000 

III 

0.4345 

0.4119 

0.4i538 

0.4095 

0.4119 

IV 

0.5554 

0.5143 

0.5411 

0.5321 

0.5607 

V 

0.5929 

0.5911 

0.5643 

0.5821 

0.5696 

VI 

0.4696 

0.4964 

0.5250 

0.4643 

0.4536 

VII 

0.4589 

0.4554 

0.4411 

0.5018  . 

0.4732 

TABLE  26. 


Subject 


8  V/mi      V 
10  ^K—-P) 


I 

840 

202773 

i 

!    0.5374 

0.0101 

0.4626 

II 

840 

146678 

0.5017 

0.0085 

0.4983 

III 

840 

45255 

1    0.4183 

0.0071 

0.5817 

IV 

560 

138717 

0.5407 

0.0083 

0.4593 

V 

560 

64868 

0.5S00 

0.0057 

0.4200 

VI 

560 

332973 

0.4818 

0.0129 

0.5182 

VII 

560 

211623 

0.4661 

0.0103 

0.5339 

TABLE  27. 
Coefficient  of  Divergence  and  Component  op  Physical  Variation. 


Subject 


I 

1.462 

0.020126 

II 

1.408 

0.017119 

III 

1.145 

0.009507 

IV 

1.290 

0.016641 

V 

1.139 

0.011380 

VI 

1..578 

0.025782 

VII 

1.396 

0.020554 

Average  of  Q  =  1.345 


186 


PROBLEMS  OF  PSYCHOPHYSICS 


M  C3  CC  O  CC  W  O 

CC  CO  M  O  CO  CO  o 

05  O)  CO  CO  «D  ^  00 

05  05  OS  00  >C  (N  rH 

O  O  O  O  O  O  O 


t^  t-  t-  O  t-  t--  o 

to  CD  CO  C  CO  iC  O 

O  O  CD  t~  CO  O  OJ 

O  O  O  "-I  •^.  f^  *? 

o  o  o  o  o  o  o 


^,  ,-  CO  -I  sc  O  -H 

05  05  05  00  o  oi  T-; 
d,  d  d>  o  d  o  o 


C-l  CO  CC  C-l  "*  CO 

C  C;  '-'  '1'.  '^  <*. 

d  o  o  d  o  c3 


CO  i>  O  r-  C  Q  t;- 

00  CO  C  «  O  O  CO 

S  it  ^  00  ci  CO  in 

05  C5  00  ■*  CO  '-^  P 

d  c  d  d  d  o  o 


f,  CO  O  CO  C  O  CO 

CO  CO  O  CO  o  p  CO 

M  CI  CO  "-I  cc  t^  3 

c  d'  -<  lO  p  00  p 

d  d  d  d  d  d  o 


t,  CO  CO  CO  o  t--  o 

«5  ■"  '^  =2  S  2  S 

05  St.  cc  p  <^  ^  P 

d  a  d  d  d  a  o 


CO  t-  t; 

g5  g  2  55  5  -35  CO 

§  q  ^  CO  CD  CO  p 

d  d  d  di  d  o  o 


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O  S  03  I-  «  CO  M 

O  t-  N  t~  ^  2  S 

o  C5  05  t^  in  --1  P 

rH  d  d  6  o  c  o 


o  (M  1^  N  t^  S  12 

O  p  p  IM  •*  CO  p 

d  d  6  d  d  d  o 


00  CO  2  I-  :i  2!  S 

t^  m  00  CD  -^  3^  S 

L  t--  00  o  t^  00  g; 

05  S  00  t-  ri;  --;  p 

d  d  d  d  o  o  o 


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d  d  d  d  d  o  o 


S  g  ::  e::  So  o  § 

05  05  05  t-  •"  "^  P 

d  d  d  d  o  o  o 


ri  C  C5  IM  CO  CO  O 

%\    C  *  <M  "  S  2 

r^  rl  GO  C-l  '^  "^  ^ 

o  p  o  CI  'j;  00  p 

d  d  d  d  d  o  o 


T)i  00  ci  CO  c  2^  50 

00  X  o;  o  s  —  — 


r-.  CO  CO  o  r^  00  05 

CO  CO  o  CO  CO  lo  »ra 

o  o  CO  "0  CO  N  00 

o  o  o  "-;  CO  CO  p 

d  d  d  d  d  d  o 


w  Cl  CD  CO  '-<  _  - 
O  O  O  -^  CO  CO  p 

d  d  d  jd  d  d  d 


t,  t^  ^  05  <M  C-l  CO 

CO  O  d  05  "*  CO  lO 

CI  <N  in  O  CD  o  -^ 

o  O  -^  ■*<>'.  '^.  p 

d  d  d  d  d  d  d 


CO  o  05  >o  >2  •;;  S 
CO  CO  o  o  s  5S  S 


d  d  d  d  d  o  o 


o  ■*■*■*  o  2 


o  d  o  ci  CO  CO 

d  d  d  d  d  d  d 


r-i  ^1  O  "^  '^  '-'^  "^ 

d  o  ^  ^'  CO  CI  p 
d  d  d  d  d  d  o 


rji     Ci    X    ^    1^ 
O  — <  CI  CO  p 

d  d  d  ■^  d 


APPENDIX 


187 


S  Q 


■y) 

Tf 

to  CO  Cl  Cl 

CI 

o 

M 

M  CO  o  c: 

I-- 

o: 

> 

to 

':o 

•O  CO  to  £K 

t^ 

o 

lO 

lO 

O  O  CO  00 

CI 

t- 

o 

o 

CO  "5  CO  CO 

ffl 

00 

.-1  CO  CO 

t/l 

rn 

00  CO  ■*  •* 

CO 

o 

CI 

X  t-<  -H  •* 

> 

CO 

Tf 

•*  00  •*  CO 

05 

o 

lO 

o 

CO  CJ  o  ^ 

•* 

CI 

o 

IM 

lO  to  rH  CO 

o 

05 

1-1  CO  CO 

00 

CO 

CI  •*  t^  00 

■<f 

Oi 

(M 

CO  o  ■*  o 

M 

CO 

> 

•* 

02  lO  CI  lO 

■o 

IN 

05 

O  CO  ■*  CI 

CO 

o 

CI 

'~' 

CO  02  CO  »— ' 

•-1  CO  N  1-1 

■^ 

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fN 

o 

00  O  U5  ■* 

CI 

r^ 

00 

N   CJ   T-l   Tl< 

CO 

a 

> 

lO 

on 

r~ 

t- 

35 

Cl 

CI  t^  05  CO 

•>»« 

Ci 

'^ 

CI 

O  00  CI  05 
^  CI  CO  ^ 

CI 

1^ 

o 

^ 

T)< 

00  ■*  o  o 

on 

■* 

1^ 

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■* 

CO 

Ttl 

»  CO  o  CO 

t—t 

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1—1 

CO  CO  CI  to 

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■M 

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■* 

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to 

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00 

o 

(-H 

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o 

CO 

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CI 

C5  CO  —  CO 
C)  CO  CI 

lO 

00 

03 

or 

o 

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N 

N 

Tt< 

o 

•*  00  — 1  to 

o 

■* 

r/1 

CO 

C2  O  ■*  CD 

o 

CO 

>— 4 

t^ 

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CO 

t~ 

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05 

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on 

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Ph 


CI  CI  CI  to  O  00  CO 

to 

lO  -<  -H  >0  O  CI  t^ 

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q 

> 

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^ 

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o 

lO  ■*  CO  lO  o 

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OS 

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to 

lO  O  OS  CO  o  t^  o 

r^ 

r^  CD  00  CO  ■*  I-  •* 

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> 

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CO 

■*  I-  -H  CD  C  ■*  CO 

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T-l  CO  CO  --1 

OS 

CI  00  '■*•<*  C  CI  CI 

CJ 

O  CZ)  Tf  X  o  •-<  o 

w 

o  00  i~  CO  i^  c;  iO 

> 

CO  I^  CO  CD  -^  CI  •* 

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d 

00  t^  X  t~  •*  1^  I^ 

CI 

i-H  .-1  CI  I-  CD  '-'  "-1 

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l-H  CO  CI  -H 

OS 

X  O  CO  O  C  CO  CO 

to 

■*  ■*  1-  CI  C:  t-  O 

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»-H 

> 

■*  CO  CD  --I . --'  X  ■-; 

t^ 

■*  .-^  X  d  ic  ci  t-^ 

00 

CO  O  CO  CO  OS  —1  CO 

CO 

-H  CI  OS  I~  CI  O  CI 

to 

CI  CO  CI 

OS 

O  CI  CO  ■*  O  C  -"l" 

to 

O  --1  >H  CI  C  X  X 

O  t^  C  O  C  CI  OS 

o 

(_H 

o  OS  ■*  -1  CO  I-  CO 

OS 

►— ' 

d  X  I-  CD  d  ^  ■*' 

00 

O  X  X  lO  CI  o  ■* 

OS 

.-1  lO  X  CO  CO  •* 

OS 

1-1  CO  CO 

OS 

CI  CI  O  CI  O  N  Ttl 

N 

CO  t^  •«*'  OS  o  CO  •* 

rJH  O  X  »-<  O  •*  t^ 

(^ 

(— t 

CO  CO  ■-;  p  i-|  00  CO 

'I' 

f— 1 

d  •*  1^  ci  d  OS  cs 

00 

O  X  OS  OS  CO  X  •* 

CO 

.-1  T-1  X  CI  .->  ■<1'  LO 

t^ 

CI  CO  CI 

OS 

CI  O  CO  O  O  CO  CO 

o 

CO  O  -<  X  O  i-O  CO 

CI 

CI  X  CI  CD  O  O  OS 

OS 

in  X  in  t^  •-;  "-^  CO 

CI 

►^ 

lo  •*'  d  •*  'I'  00  i-o 

CI 

^  in  .-1  CI  CD  CO  CO 

X 

^  OS  X  X  c:s  -"i" 

^  CI  CO 

o 

■>!f<  X  CI  to  O  ■«1"  00 

^ 

X  X  OS  OS  c  o  o 

»-t  »-l  t-t 

188 


PROBLEMS  OF  PS  YCHOPHYSICS 


TABLE  32. 
Observations  on  the  Just  Perceptible  Positive  Difference.     SUBJECT  L 


IVa 

I 

III 

IV 

Tk 

Nk 

TkNk 

Nk 

TkNk 

Nk  : 

TkNk 

Nk 

l-kNk 

84 















88 

— 

3 

264 

1 

88 

3 

264 

92 

9 

828 

8 

736 

7 

644 

6 

552 

96 

16 

1536 

24 

2304 

20 

1920 

16 

1536 

100 

15 

1500 

15 

1500 

36 

3600 

40 

4000 

104 

58 

6032 

43 

4472 

35 

3640 

33 

3432 

108 

2 

216 

7 

756 

1 

108 

2 

216 

I 

100 

10112 

100 

10032 

100 

10000 

100 

10000 

Average 

101.12 

100.32 

100.00 

100.00 

TABLE  33. 
Observations  on  the  Just  Perceptible  Positive  Difference.     SUBJECT  II. 


Nk 

[Va 
l-k  Nk 

I 

III 

IV 

fk 

Nk 

rkNk 

Nk 

rtNk 

Nk 

TkNk 

84 

4 

336 

2 

168 

1 

84 

1 

84 

88 

— 

3 

264 

4 

352 

3 

.  264 

92 

8 

736 

18 

1656 

7 

644 

12 

1104 

96 

25 

2400 

26 

2496 

27 

2592 

23 

2208 

100 

30 

3000 

23 

2300 

28 

2800 

37 

3700 

104 

24 

2496 

19 

1976 

31 

3224 

21 

2184 

108 

8 

864 

6 

648 

2 

216 

2 

216 

I 

99 

9832 

97 

9508 

100 

9912 

99 

9760 

Average 

99.31 

98.02 

99.12 

98.59 

APPENDIX 


189 


TABLE  34. 
Observations  on  thb  Just  Perceptible  Positive  Difference.     SUBJECT  III. 


IVa 

I 

III 

IV 

Tfc 

Nk 

TfeNfe 

Nk 

TkNk 

Nk 

TkNk 

Nk 

TkNk 

84 



1 

84 







88 

4 

352 

3 

264 

— 

1 

88 

92 

10 

920 

19 

1748 

4 

368 

3 

276 

96 

21 

2016 

36 

3456 

8 

768 

20 

1920 

100 

36 

3600 

34 

3400 

32 

3200 

41 

4100 

104 

28 

2912 

5 

520 

56 

5824 

30 

3120 

108 

1 

108 

2 

216 

— 

5 

.  540 

I 

100 

9908 

100 

9688 

100 

10160 

100 

10044 

Average 

99.08 

96.88 

101.60 

100.44 

TABLE  35. 
Observations  on  the  Just  Perceptible  Positive  Difference.     SUBJECT  IV. 


I 

III 

IV 

kk 

Nk 

TkNk 

Nk 

TkNk 

Nk 

TkNk 

84 



2 

168 

4 

336 

88 

2 

176 

3 

264 

1 

88 

92 

8 

736 

20 

1840 

4 

368 

96 

35 

3360 

28 

2688 

27 

2592 

100 

32 

3200 

24 

2400 

34 

3400 

104 

22 

2288 

20 

2080 

25 

2600 

108 

1 

108 

3 

324 

5 

540 

I 

100 

9868 

100 

9764 

100 

9924 

Average 

98.68 

97.64 

99.24 

190 


PROBLEMS   OF    PSYCHOPH YSICS 


TABLE  36. 
Observations  on  the  Just  Perceptible  Positive  Difference.     SUBJECT  V. 


I 

III 

IV 

Tk 

Nk 

TkNi 

Nk 

TkNk 

Nk 

rkNk 

84 

2 

168 

1 

84 



88 

1 

88 

— 

3 

264 

92 

9 

828 

16 

-     1472 

13 

1196 

96 

29 

2784 

36 

3456 

63 

6048 

100 

30 

3000 

35 

3500 

17 

1700 

104 

19 

1976 

12 

1248 

4 

416 

108 

10 

1080 

— 

— 

I 

100 

9924 

100 

9760 

100 

9624 

Average 

99.24 

97.60 

96.24 

TABLE  37. 
Observations  on  the  Just  Perceptible  Positive  Difference.     SUBJECT  VI. 


I 

III 

IV 

fk 

Nk 

rkNfe 

Nk 

TkNk 

Nk 

rkNk 

84 

2 

168 







— 

88 

6 

528 

— 

— 

1 

88 

92 

13 

1196 

5 

460 

2 

184 

96 

25 

2400 

8 

768 

18 

1728 

100 

•    31 

3100 

44 

4400 

20 

2000 

104 

21 

2184 

37 

3848 

32 

3328 

108 

1 

108 

6 

648 

23 

2484 

I 

99 

9684 

100 

10124 

96 

9812 

Average 

97.82 

101.24 

102.21 

APPENDIX 


101 


TABLE  38. 
Observations  on  the  Just  Perceptible  Positive  Difference.     SUBJECT   VII. 


I 

III 

IV 

Tk 

Nk 

TfcNk 

Nk 

TfcNk 

Nk 

TkNk 

84 

1 

84 

1 

84 

_ 

_ 

88 

1 

88 

1 

88 

— 

— 

92 

7 

644 

4 

368 

8 

736 

96 

6 

576 

18 

1728 

19 

1824 

100 

35 

3500 

27 

2700 

39 

3900 

104 

31 

3224 

38 

3952 

30 

3120 

108 

19 

2052 

7 

756 

4 

432 

I 

100 

10168 

96 

9676 

100 

10012 

Average 

101.68 

100.79 

100.12 

TABLE  39. 

Differences  between  Observed  and  Calculated  Values  op  the  Just  Perceptible 
Positive  Difference. 


Subject 

Differences  between  observed  and  calculated  values. 

1 

, 

Calculated  value 

IVa 

I                     III 

IV 

ToUl 

I 

99.76 

+  1.36 

+  0.56 

+  0.24 

+  0.24 

+  0.60 

II 

98.31 

+  1.01 

-0.29 

+  0.81 

+  0.28 

+  0.40 

III 

99.82 

-0.74 

-0.94 

+  1.78 

+  0.62 

-0.32 

IV 

97.98 

+  0.70      j          -0.34 

+  1.26 

+  0.54 

V 

96.01 

+  2.33 

+  0.69 

-0.67 

+  0.78 

VI 

99.21 

-1.39 

+  2.03 

+  3.00 

+  1.20 

VII 

98.76 

+  2.92 

+  2.03 

+  1.36 

+  2.17 

192 


PROBLEMS  OF  PSYCHOPHYSICS 


> 

z 

QO  (O  «  00  O  O  O 
to  t^  ■*  C^  O  Ol  •* 

,1  ^  t^  ^  — I  (M  CI 

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to 
>o 

<Xj 
OS 
CJ 

to 

X 

d 
o 

M 
Z 

(N  CJ  CS  CO  -H  Oi  O 
rl  ■*  O  0>  CO 

1 

> 

A! 
M 

00  to  o  to  o  o  o 

to  -^  ti  en  o  CD  -"i" 

.-(  o  CO  00  lO  CO  c-i 

n  Tji  o  a>  CO 

o 
to 

CS 

o 

1 

M 
Z 

C)  t-  O  .1  lO  o  o 
C)  10  O)  OJ  CO 

cq 

> 

(N  (N  to  00  O  O  O 

lO  lO  05  GO  O  ^  00 

Cq  CO  •*  CI  C5  to  O 

CO  O  00  CO  'H 

00 

o 
CO 

C5 
CI 

to 

05 

q 

05 

z 

CO  ■*  »  CX3  C^  "5  O 
CO  IN  X  CO  -" 

o 
o 
CO 

> 

z 

•«J<  00  -^  O  O  00  N 
O  IM  -^  ■>*  O  to  t^ 
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C^  00  05  CO 

to 
>o 

05 

•o 

00 

Z 

CD  te  CI  o  o  f~  05 
CO  05  03  to 

o 
o 

CO 

S 

M 
Z 

li 

■^  •*  CI  o  o  to  •<1< 

00  O  ^^  CO  o  t^  to 

t-  CO  -^  CO  CO  00 

CO  00  '!<  C) 

o 
o 

00 

OS 
CO 

05 

Z 

,-1  X  to  lO  CO  C5  X 
CO  00  ■>*  'H 

o 
o 

a 

z 

li 

O  O  O  CD  O  O  •* 

t-  00  •*  C5  o  oo  2; 

to  CO  — '  O  CO  00  05 
t3<  05  -"I  05  ^ 

CJ 

s 

05 
CO 

00 
05 

z 

00  O  >0  ^  00  lO  00 

.-1  ■*   O  -^  05  rH 

lO 

05 

CO 

« 

z 

li 

to  O  CO  O  CD  to 

1   ^  to  C)  O  t-  O) 

CD  1^  CI  CO  lO  CJ 

C)  I^  O  t-  "^ 

o 

o 
to 
CO 
d 

o 

z 

t^  O  CD  to  C5  CI 

1     CO  t-  O  to  f-H 

§ 

T)<  00  CI  to  O  ■*  00 

00  00  05  Ci  o  o  o 

^ 

4) 

> 

< 

> 

O  lO  lO  C5  O  — 1  o 
O  OS  O  1"  CO  CO  o 
00  OS  ■*  CJ  Tf  o  o 

rH  -1  CO  CI  q  q  q 
d>  o  d  d  o  d  d 

o  o 
o  o 
o  o 
o  o 
-i  d 

> 

t-  1^  00  CO  CI  CI  -« 

to  to  O  ■*  CO  CO  o 
^  C)  X  CI  ■*  o  o 
.I  c)  CO  c>  q  q  q 
d  d  d  d  d  d  d 

o  o 
o  o 
o  o 
o  o 
-i  d 

> 

f,  to  to  to  to  00  o 
to  CI  d  ^  o  -^  --< 
lO  d  to  t^  •*  ■*  o 
o  --I  -CI  c)  CI  q  q 
d  d  d  d  d  d  d 

OS  o 
OS  o 
OS  o 
OS  O 

d  d 

> 

o  LO  1*  ■*  05  CO  in 

O  QO  O  CI  t^  O  O 
t^  O  CI  C»  lO  CI  o 

o  -<  CO  CO  ■-;  q  q 
d  d  d  d  d  d  d 

o  o 
o  o 
o  o 
o  o 
^    d 

5 

C5  N  lO  lO  CJ  Cl  Cl 

CI  CJ  CI  CO  CO  to  o 
•<}<  o  i-O  --I  X  o  o 

o  ^  ■*  CO  o  q  q 
d  d  d  d  d  d  d 

o  o 
o  o 
o  o 
o  o 
^  d 

« 

0.0956 
0.1668 
0.3475 
0.2757 
0.1017 
0.0124 
0.0003 

o  o 
o  o 
o  o 
o  o 
^  d 

- 

O  -^  05  CO  •*  o  o 
O  X  CO  O  O  f-  o 
CO  05  05  t-  r^  o  o 
o  o  •*  CI  o  q  q 

d  d  d  d  d  d  d 

§  § 
o  o 
o  q 
«  d 

X  -"J"  O  CO  CI  X  1" 

O  O  O  05  OS  X  X 

i 

APPENDIX 


193 


> 

19.4400 

20.7480 

34.9520 

21.5904 

3.9560 

0.2728 

0.0000 

c» 

"5 

d 

o 

i  > 

to  00  O  00  ■*  to  ■^ 
CO  to  O  CI  •*  -1  00 

o  r-  30  CO  CO  »  o 
to  o  q  lO  'i;  CI  q 
ci  co'  00  -H  -^r  d  d 

•^  M  PO  c^ 

to 

lO 

d 

o 

i 

> 

to  •>*  O  CO  Cl  1-  o 
CO  C  — '  CO  iO  C)  ■* 

c)  lo  to  r~  CO  1"  00 
^  t^  ci  q  -H  q  q 
to  ci  to  to  ci  CO  d 

^  <N  Cl  CI 

Cl 

o 

CO 

> 

i 

7.5600 
11.2840 
32.0370 
30.9504 
14.5268 
1.7864 
0.0420 

to 

to 

00 

00 
05 

1 
1 

1 

4.5576 

10.6288 

45.2530 

30.0960 

7.6544 

0.5456 

0.0168 

CJ 
CJ 
"O 

00 

OS 

03  CI  O  Cl  'J'  Cl  CJ 
•*  t^  CI  1^  to  '^  lO 

cj  Ti<  in  to  i-o  c:  CI 
CO  CO  t-  ■*  CO  q  q 
d  t^  ■«)<  td  d  '-'  d 
^  r^  n  c-i 

o 
to 

CO 

d 

05 

6.4800 

10.2024 

49.3860 

25.9776 

6.4768 

0.6160 

0.0000 

00 

00 

CO 

d 

OJ 

i 

00  •<*i  O  to  CJ  00  '1< 
O  O  O  C5  »  00  00 

^} 

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X 

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X 

CI  en  o  r^  lo  3  o 

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in  r-  CI  to  o>  o  o 

•-• 

^ 

OS  t-  m  CI  CO  •«»'  o 

CO 

> 

o>  m  05  r-  to  c» 

O  -H  ■*  O  CO 

Cl  Cl  CO  Cl 

"^ 

00  (M  O  00  00  00  to 

o 

X  r^  o  oo  ■<)<  o  o 

X  X  o  ■*  to  X  o 

--  o;  o  -■  o)  t-  t- 

t- 

> 

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,-1 

to  m  o  CO  o  c) 

CI 

CO  ■<><  X  O  tH 

-H  Cl  CO  CI 

00  to  o  to  '»<  Cl  o 

CD 

X  -H  o  m  X  ^  to 

■*  ^  o  to  CO  CO  in 

> 

CO  c  -■  C  't  o  o 

05 

-H  CO  to  CO  o  to  r- 

o 

CD  Cl  CI  C  CO  'fi 

o 

CD  CO  o  in  o  CO 

o 

rt  CI  CI  CI 

05 

O  O  O  ■*  CO  CI  o 

c< 

o  to  o  X  in  CO  X 

X  CO  O  CO  'to  O  CI 

lO 

> 

■^  I.-  1-  CI  ■*  CI  in 

t— « 

O  CO  CO  -<  CO  t^  CO 

CI 

— 1  t-  o  w  CO  m 

to 

X  -•  CI  O)  CO  ^ 

o 

--^  CO  CI  ^ 

05 

X  CI  O  O  X  X  CI 

X 

o  in  o  to  ^  CI  -H 

o 

CI  o>  o  —  O  -<  -H 

H-t 

CI  CO  CO  CI  Cl  o  ■* 

t^ 

^ 

CI  in  m  o  ■*  X  i-H 

in 

0-.  O  Cl  X  o  ■«»< 

to 

■>!»<  -H  O  X   t- 

rt  ■q'  CI 

O) 

■«1<  X  O  CJ  X  to  00 

to 

X  X  o  -^  X  in  CD 

05 

r^  o  o  i-n  X  CI  -< 

to 

1— ( 

O  -H  CI  X  1-  o  — 

t— ( 

in  •^  m  c  o  CO  CI 

•* 

•^  O  t-  Tf  O  35 

05 

-H  X  t  O  X 

-H  ^  CO  CI 

05 

o  to  o  to  to  o  o 

00 

O  C5  O  (35  lO  00  O 

M 

■*  Tj<  o  ■*  to  o  o 

ti  1 

X  O  to  X  X  CI  o 

•* 

05  — •  X  CO  in  •*  o 

CO 

OS  to  CO  05  C5  in 

If 

to  o  05  •*  in 

» 

^  ■*  CI 

05 

^ 

X  ■^  O  to  Cl  OO  Tl< 

^ 

u 

O  O  O  05  05  00  X 

>-H  ,-1  rf                    1 

194 


PROBLEMS  OF  PSYCHOPH YSICS 


TABLE  44. 
Observations  on    the   Just   Imperceptible    Positive  Difference.     SUBJECT  I. 


IVa 

I 

III 

IV 

fk 

Nk 

TfeNk 

Nk 

TkNfc 

Nk 

Nkfk 

Nk 

rkNk 

84 

88 

5 

440 

2 

176 

1 

88 

92 

9 

828 

12 

1104 

9 

828 

6 

552 

96 

15 

1440 

13 

1248 

31 

2976 

40 

3840 

100 

58 

5800 

56 

5600 

44 

4400 

42 

4200 

104 

5 

520 

11 

1144 

11 

1144 

8 

832 

108 

8 

864 

6 

648 

5 

540 

3 

324 

I 

100 

9892 

100 

9920 

100 

9888 

100 

9836 

A-Verage 

98.92 

99.20 

98.88 

98.39 

TABLE  45. 
Observations  on  the  Just  Imperceptible  Positive  Difference.     SUBJECT  II. 


IVa 

I 

III 

[V 

Tk 

Nk 

i-kNk 

Nk 

TkNfc 

Nk 

l-kNK 

Nk 

TkNk 

84 

1 

84 

88 

1 

88 

4 

352 

2 

176 

3 

264 

92 

13 

1196 

12 

1104 

11 

1012 

14 

1288 

96 

27 

2592 

20 

1920 

33 

3168 

38 

3648 

100 

28 

2800 

35 

3500 

45 

4500 

25 

2500 

104 

17 

1768 

19 

1976 

5 

520 

12 

1248 

108 

14 

1512 

10 

1080 

4 

432 

7 

756 

I 

100 

9956 

100 

9932 

100 

9808 

100 

9788 

Average 

99.56 

99.32 

98.08 

97.88 

APPENDIX 


195 


TABLE  46. 
Observations  on  the  Just  Imperceptible  Positive  Difference.     SUBJECT  III. 


IVa 

I 

III 

IV 

fk 

■Nk 

TfcNk 

Nk 

TfcNk' 

Nk 

TkNk 

Nk 

TkNk 

84 

88 

92 

96 

100 

104 

108 

2 
10 
33 
45 
10 

176 

920 

3168 

4500 

1040 

2 
29 
48 
14 

7 

184 
2784 
4800 
1456 

756 

4 
17 
43 
36 

368 
1632 
4300 
3744 

1 

5 

IS 

40 

31 

5 

88 

460 

1728 

4000 

3224 

540 

I 

100 

9804 

100 

9980 

100 

10044 

100 

10040 

Average 

98.04 

99.80 

100.44 

100.40 

TABLE  47. 
Observations  on  the  Just  Imperceptible  Positive  Difference.     SUBJECT  IV. 


I 

[II 

IV 

Tk 

Nk 

TfcNk 

Nk 

TkNk 

Nk 

Nkk 

84 

1 

84 

88 

2 

176 

5 

440 

2 

176 

92 

18 

1656 

19 

1748 

18 

1656 

96 

27 

2592 

25 

240 

28 

2688 

100 

38 

3800 

29 

2900 

36 

3600 

104 

S 

832 

12 

1248 

13 

1352 

108 

7 

756 

9 

972 

2 

216 

I 

100 

9812 

100 

9792 

99 

9688 

Average 

98.12 

97.92 

97.86 

196 


PROBLEMS  OF  PSYCHOPHYSICS 


TABLE  48. 
Observations  on  the  Just  Imperceptible  Positive  Difference.     SUBJECT  V. 


[ 

III 

IV 

fk 

Nk 

TkNk 

Nk 

TkNk 

Nk 

I-fcNk 

84 

88 

4 

352 

8 

704 

92 

8 

736 

19 

1748 

59 

5428 

96 

24 

2304 

39 

3744 

20 

1920 

100 

30 

3000 

28 

2800 

10 

1000 

104 

28 

2912 

6 

624 

2 

208 

108 

10 

1080 

4 

432 

1      , 

108 

I 

100 

10032 

100 

9700 

100 

9368 

Average 

100.32 

97.00 

93.68 

TABLE  49. 
Observations  on  the  Just  Imperceptible  Positive  Difference.     Subject  VI. 


I 

III 

IV 

l-k 

Nk 

TkNk 

Nk 

TkNk 

Nk 

TkNk 

84 

. 

88 

2 

176 

92 

12 

1104 

2 

184 

4 

368 

96 

25 

2400 

32 

3072 

12 

1152 

100 

36 

3600 

44 

4400 

33 

3300 

104 

14 

1456 

16 

1664 

36 

3744 

108 

11 

1188 

6 

648 

15 

1620 

I 

100 

9924 

100 

9968 

100 

10184 

Average 

99.24 

99.68 

101.84 

APPENDIX 


197 


TABLE  50. 
Observations  on  the  Just  Imperceptible  Positive  Difference.     SUBJECT    VII. 


I 

III 

IV 

fk 

Nk 

I-kNk 

Nk 

TkNk 

Nk 

J-kNk 

84 

1 

84 

1 

84 

88 

4 

352 

4 

352 

1 

88 

92 

14 

1288 

9 

828 

16 

1472 

96 

21 

2016 

37 

3552 

24 

2304 

100 

33 

3300 

28 

2800 

42 

4200 

104 

22 

2288 

17 

1768 

16 

1664 

108 

5 

540 

3 

324 

1 

108 

y 

100 

9868 

99 

9708 

100 

9836 

Average         ; 

98.68 

;             98.06 

1 

98.36 

TABLE  51. 
Observed  Values  of  Just  Perceptible  and  Just  Imperceptiblb  Positive  Diffbrencb. 


Subject. 

•IVa 

I 

III 

IV 

A 

B 

A 

B 

A 

B 

A 

B 

I 

II 

III 

IV 

V 

IV 

VI, 

101.12 
99.47 
99.08 

98.92 
99.56 
98.04 

100.32 
98.02 
96.88 
98.68 
99.24 
97.82 

101.68 

99.20 
99.32 
99.80 
98.12 
100.32 
99.24 
98.68 

100.00 
99.12 

101.60 
97.64 
97.60 

101.24 

100.79 

98.88          100.00 
98.08           98.59 
100.44   i      100.44 
97.92   1        99.24 
97.00   i        96.24 
99.68         102.21 
98.06          100.12 

98.36 
97.88 

100.40 
97.86 
93.68 

101.84 
98.36 

198 


PROBLEMS  OF  PSYCHOPHYSICS 


00    W    »    C^    O    O    IM 

CI 

■X 

to  o»  00  r-  o  CI  1- 

CD 

2; 

rH     t^    lO     M    CO     t^    OJ 

■* 

CO 

CO     I-    O    lO 

X 

'"' 

> 

M 
Z 

(M    05    0»    M    CO    in    05 

05 

CO    00    O    lO 

c» 

m    CO   •*    o   •*   5D 

CO 

CO 

2 

I-    1.0    CI    O    CO    ".O 

rt     CO     CO    CO     M    'l- 

o 

I 

1 

.-H    CO     ^    CO    CO 

CO 

o 
o 

H.4 

■            > 

i! 

(M  »  a;  CO  CO  IN 

^ 

r-  CO  -^  CO  CO 

2 

CO 

CO   CI   00   o   ■^   o 

o 

o 

Z 

1^    — 1    CD    O    't    CI 

C    OS    Oi    OC    t^    CO 

1—1 

o 

1 

rH      t~     t^     CD     CO     -H 

o 

05 

> 

M 

CI    CD    CO    OC    CD    Lt 

o 

2 

CO 

2 

■*    CI    O    O    O    IM    •* 

CI 

t~ 

00    O:    CD    00    O    CO    ■* 

t^    O    CD    CO    ■^    O 

CI 

o 

> 

.M 

^    0>    >0    O    CO    CO    00 

05 

2 

lO    «    O    CO    rt 

^i 

,J>! 

■*    CI    CI    O    ■*    CO 

cc 

o 

2 

O    CO    -•    O    CD    Oi 

CD 

CI    O    CO    CD    ■*    C) 

.14 

r^     C2     t^     O     ■-< 

OS 

Vi 

»—< 

1 

M 

CO  -"i  t-  CO  -<  CI 

o 

2 

Tt< 

.14 

■*   O   O   CC   O    CI   o 

■* 

o 

2 

00  00   O   CI   o   --^   cc 

.1^ 

^  rt  CO  "C  CO 

05 

X 
03 

u 

■-H      rH 

CO 

1— t 

1 

■^ 

-H    O    O    OC    CO   CO   >o 

o 

—  LO  -•  CO  i^'  CO 

2 

'J' 

j< 

Tt<    CI    ■*    O    O   CO 

CO 

o 

2 

o  ^  c  o  ■*  r- 

CO 

>o 

- 

1- 

CO    -    3    CO    C. 

05 

CO 

X 

o> 

X    to    Oi    O    lO    CI 

o 

.£<: 

CO    O    O    CO    C4 

•* 

2 

<D 

bo 

ea 

1 

■*    X    CI    CO    O    'l'    X 
»   X   05   C5   o   o   o 

^ 

0) 

> 

< 

> 

o 

c 

! 

O   CO   O   CO   CO   o   o 
O   CO   O   CO   CO    o   o 
CI    O    O    ■*    CO    ■*    CO 
O    ^    CI     O    r-;    CJ    05 

d  di  d>  d  o  d>  o 

- 

O   t-   O   I-   I-   o   o 

O  CO  o  o  o  o  c 

X    C5    ^    lO    CO    CD    ■* 
05   X   t^   ■*   CI   o   o 

d  d  d  d  d  d  d 

> 

o 

t^    t~   CO    O    CO    CO    o 

CD    CO    CO    O    CO    CO    O 
CI    CO    X    -1    C5    CO    X 
O    ^    CI    lO    1-;    05    O! 

d  d  d  d  d  d  d 

CO  CO  t-  o  r^  r-  O 
CO    CO    O    O    CO    CO    o 
t~    CO    — '    05    O    CO    CI 
C5   X    t-   r)<    CI    o   o 

d  d  d  d  d  d  d 

> 

o 
c 

O   CO  O   O  CO   t^  o 
O    CO   O   O   CO   CO   o 
■*    1^    O    -H    CI    CO    X 
O    O    CO     1--;    00    05    05 

d  d  d  d  d  d  d 

-■ 

o  r-  o  o  t-  CO  o 

O    CD    O    O    CO    CO    O 
CD    CI    O    C5    I^    CO    CJ 

o;  q  t-  c^  —  q  q 

d  d  d  d  d  d  d 

> 

_,        ot^coocoo 
,■          O   CO  CO   o   o   o  c 
■S        i<f^cococ;iox 
g        o  o  CI  lO  t-;  05  cr. 

d  d  d  d  d  d  d 

- 

O  CO  t-  o  c  o  o      ! 

O    CO    CD    O    O    C   O 
CO    CI    CD    -*    --^    lO    CI 

<s  C5  t-  'j;  CI  q  q 
d  d  d  d  d  d  d 

t— t 

o 

•*    Tjt    ,-<    O    O    O   CO 

Tt<    ■*    ^    O    O    O   CO 
O    ■*    CO    O    CI    •*    t^ 

o  o  --I  CO  q  q  q 
d  d  d  d  d  d  d 

- 

CD  to  05  o  o  g  t- 

in   lO   X   O   O   O  CO 
O    O    CD    O    X    CO    CJ 

05  05  X  w  CO  q  q 
d  d  d  <z>  d  d  d 

t^ 

o 

0.0667 
0.1378 
0.3000 
0.5511 
0.7689 
0.9044 
0.9844 

- 

CO    Cl    O    05    -^    «0    CD 
CO    CI    O    X    ■-    >«    "3 

CO     CO     O     "^     (^     05     •-1 

S  X  S  ^  CI  q  q 
d  d  d  d  d  <z>  <^ 

o 

c 

Tj<  T)<  o  •*  X  t-  M 
•*    •*    O    ■*    t~    CD    CO 

CD    -^    CI     CO    1^    X    05 

o  --<  ■*  'C  00  q  ® 
d  d  d  d  d  d  d 

- 

CD   CO  O   CO   Cl   CO   t~ 
lO   »o   O    *0    Cl    CO   CD 

CO  ira  X  CO  CI  -:<  o 
05  00  lO  CO  ■-;  q  q 
d  d  d  d  d  d  d 

1     Tj-    X    CI    O    O    3"    00 

X    X    O:    C-.    C    O    O 

APPENDIX 


19  9 


TABLE  54. 

Values  op  the  U's  for  the  Determinatiom  of  the  Just  Perceptible  Nbgativb 
Difference. 


I 

II 

III 

IV- 

V 

VI 

VII 

108 

0.0067 

0.0156 

0.0267 

0.0200 

0.0200 

0.0200 

0.0400 

104 

0.0132 

0.0941 

0.0584 

0.0490 

0.0327 

0.0653 

0.0576 

100 

0.1198 

0.2058 

0.3477 

0.1955 

0.1674 

0.1890 

0.2406 

96 

0.2887 

0.3073 

0.3971 

0.3236 

0.2262 

0.3556 

0.3022 

92 

0.3315 

0.2641 

0.1479 

0.3158 

0.3876 

0.2652 

0.2661 

88 

0.2054 

0.0976 

0.0213 

0.0887 

0.1539 

0.0905 

0.0838 

84 

0.0324 

0.0145 

0.0009 

0.0071 

0.0117 

0.0139 

0.0095 

I 

0.9977 

0.9990 

1.0000 

0.9997 

0.9995 

0.9995 

0.9998 

R 

0.0022 

0.0010 

0.0000 

0.0003 

0.0005 

0.0004 

0.0002 

T^ 

lBLE  55. 

V 

ALUES   OP  T^ 

Uij  POR  the  Determination  of  the  Just  Perceptible  Negative 

Di 

fference. 

1 

I 

II 

III 

1       >v 

V 

VI 

VII 

84 

2.7216 

1.3104 

0.0756 

0.5964 

0.9828 

1.1676 

0.7980 

88 

18.0752 

8.5888 

1.8744 

7.8056 

13.5432 

7.9640 

7.3744 

92 

30.4980 

24.2972 

13.6068 

28.9436 

35.6592 

24.3984 

24.4812 

96 

27.7152 

29.5008 

38.1216 

31.0756 

21.7152 

34.1376 

29.0112 

100 

11.9785 

20.5764 

34.7685 

'     19.5509 

16.7354 

18.9030 

24.0627 

104 

1.3728 

9.7864 

6.0736 

5.0960 

3.4008 

6.7912 

5.9904 

108    , 

0.7236 

1.2180 

2.8836 

2.1600 

2.1600 

2.1600 

4.3200 

I 

93.0849 

95.2780 

97.4041 

1    95.2281 

1      94.1966 

95.5218 

96.0379 

200 


PROBLEMS  OF  PSYCHOPHYSICS 


O  ^»  •*  IN  C  IC  O 

■* 

<M  r^  o  ic  o  -^  o 

CO  •*  t^  t^  r-  C  <0 

1"! 

o  o  w  o  c<i  o  in 

»-l 

r' 

t~  CO  '^■|  in  o  CO  <r> 

2 

O  -^  lO  X  O  !M  o 

©  M  1^  •«<  «D  ■* 

(N  N  (M 

■*  O  OO  5D  g  2  O 
CO  (M  M  O  2  5^  O 

CO 

> 

CO 

.  tt  IN  N  52  t"  ^ 

o 

(N  03  '^ 

(M  i£>  -^  iM  O  N  O 

CO 

iO  ""1  CO  Ci  O  CO  o 

lO  O  ■*  lO  ■*  OD  X 

CO 

> 

lO  X  CD  CO  "^  1^  ^1 

'"* 

W  — 1  o  ■«>  CO  CO  CO 

o 

X  35  X  X  t^  lO  CO 

— 1  ra  o  to  CO  iM 

— 1  CO  c-i  ^ 

!0  00  M  5D  O  O  O 

CI 

(^  IM  rt  t^  O  rj<  O 

CO 

03  05  ^  lO  »  X  X 

> 

O  00  X  M  O  05  IN 

•* 

o  CO  IN  CO  m  05  CO 

1— ( 

us  00  CO  X  "O  C^l  CO 

o 

CD  CO  cs  o>  lO  ^^ 

(N  IN  --1 

05 

■^  IN  CO  CD  O  •*  X 

o 

O  t^  lO  CO  O  •<3<  X 

o 

lO  -^  CI  t^  lO  i-O  CI 

CO 

l_l 

CO  05  X  CO  00  CD  "* 

t^ 

s 

CO  ■*  --I  Ci  CO  --^  -^ 

CI 

CO  lo  in  t-  CO  " 

o 

>n 

-<  CO  CO 

OS 

CO  •*  •*  X  O  CO  o 

X 

CO  ■>#  CI  CO  O  lO  "9" 

CD 

t^  -^  ■*  1^  Tf  X  ■* 

t~ 

t— 1 

O  X  CO  O  CO  I^  lO 

o 

*"* 

o  »n  in  CI  t»  r-  --I 

o 

■-1  in  CO  CO  m  —1  CO 

■* 

•H  t~  CI  X  o  o  -^ 

Cl  C)  CI  -H 

05 

•'1'  CO  O  CI  O  CI  X 

CI 

■*  t^  CD  05  O  — <  X 

t^ 

-<  rH  -«  K3  in  t^  Tf 

r^ 

h-» 

CD  CD  X  CO  00  r-  -H 

■* 

00  o  in  o  f^  CJ  X 

Tt< 

IN  05  C  CO  05  ■>J<  t~ 

o 

C4  in  X  CD  -"  -^ 

r^ 

X 

•fli  X  N  CO  O  •^i  X 

1 

M  X  05  05  C  O  O 

•H  »-K  1-H 

> 

it 

Cl  X  CD  CO  O  C)  O 

•o  CO  05  r^  o  -^  c 

IN  05  Tji  OS  ^  CO  CI 

CO  c)  .-H 

c 

CI 

CO 
05 

05 

X 

2 

CO  -H  X  —1  —1  CO  o 
rt  CO  CO  -H 

05 

05 

s 

■^  Tti  CO  X  O  00  X 
X  ■*  CO  CO  o  o  o 

-H  c  -:<  CO  o  -H 
^  CO  c<3^  -H 

X 

CO 

O! 

05 

-H  M  CO  CO  CD  CI  r< 

T^  CO  CO  rt 

05 
05 

- 

z 

CI  ■^  O  X  o  •* 

in  ■^  ■*  X  o  o 

C)  CO  X  CO  o  ^ 

CO  -H  CI  rH 

X 
CI 

C) 

05 

X 

o 

ci 

05 

2 

CO  X  O  X  O  -H 
CO  CI  CJ  -H 

§ 

> 

2 

O  CD  ■*  O  O 
CI  t^  CD  CI  O 

T}(  CO  X  05  CD 

C)  CO  -H 

05 

05 

2 

in  r^  CI  o  CD 

CI  Tl'  N 

i 

•«<  X  CI  CO  O  iJ"  X 
X  X  05  05  o  o  o 

tN 

<: 

APPENDIX 


201 


TABLE  58. 
Observations  on  the  Just  Perceptible  Negative  Difference.     SUBJECT  II. 


IVa 

I 

III 

IV 

ffc 

N. 

fkNk 

Nk 

Tk'Nk 

Nk 

TfeNk 

Nk 

l-kNk 

84 

2 

168 

1 

84 

2 

168 

4 

336 

88 

6 

528 

12 

1056 

13 

1144 

17 

1496 

92 

30 

2760 

23 

2116 

28 

2576 

33 

3036 

96 

29 

2784 

29 

2784 

24 

2304 

30 

2880 

100 

21 

2100 

19 

1900 

28 

2800 

14 

1400 

104 

10 

1040 

13 

1352 

1 

104 

2 

208 

108 

2 

216 

3 

324 

4 

432 

I 

100 

9596 

100 

9616 

100 

9528 

100 

9356 

Average 

95.96 

96.16 

95.28 

93.56 

TABLE  59. 
Observations  on  the  Just  Perceptible  Negative  Difference.      SUBJECT  III. 


IVa 

I 

III 

IV 

fk 

Nk 

TkNk 

Nk 

l-kNk 

Nk 

l-kNk 

Nk 

i-kNk 

84 

2 

168 

88 

4 

352 

2 

176 

1 

88 

92 

17 

1564 

13 

1196 

9 

828 

13 

1196 

96 

38 

3648 

45 

4320 

42 

4032 

40 

3S40 

100 

34 

3400 

27 

2700 

45 

4500 

34 

3400 

104 

5 

520 

8 

832 

o 

208 

10 

1040 

108 

5 

540 

2 

216 

2 

216 

I 

100 

9652 

100 

9764 

100 

9784 

100 

9780 

Average 

96.52     ' 

1 

97.64 

97.84 

97.80 

202 


PROBLEMS  OF  PSYCHOPHYSICS 


TABLE  60. 
Observations  on  the  Just  Perceptible  Negative  Difference.     SUBJECT  IV. 


I 

[II 

IV 

l-k 

Nk 

1-kNk 

Nk 

TkNic 

Nk 

TkNk* 

84 

1 

84 

1 

84 

2 

168 

88 

15 

1320 

6 

528 

5 

440 

92 

35 

3220 

36 

3312 

37 

3404 

96 

28 

2688 

34 

3264 

22 

2112 

100 

17 

1700 

17 

1700 

24 

2400 

104 

3 

312 

5 

520 

6 

624 

108 

1 

108 

4 

432 

I 

100 

9432 

99 

9408 

100 

9580 

Average 

94.32 

95.03 

95.80 

TABLE  61. 
Observations  on  the  Just  Perceptible  Negative  Difference.     SUBJECT  V. 


I 

III 

IV 

Tk 

Nk 

i-kNk 

Nk 

TkNk 

Nk 

rfcNk 

84 

1 

84 

3 

252 

88 

10 

880 

8 

704 

27 

2376 

«2 

33 

3036 

30 

2760 

53 

4876 

-96 

31 

2976 

28 

2688 

10 

960 

100 

18 

1800 

28 

2800 

4 

400 

104 

6 

624 

3 

312 

1 

104 

108 

1 

108 

3 

324 

1 

108 

I 

100 

9508 

100 

9588 

99 

*     9076 

Average 

95.08         j 

1 

95.88 

91.68 

APPENDIX 


203 


TABLE  62. 
Observations  on  the  Just  Perceptible  Negative  Difference.     SUBJECT  VI. 


I 

III 

IV 

Tk 

Nk 

TkNk 

Nk' 

i-kNk 

Nk 

TkNk 

S4 

1 

84 

1 

84 

88 

16 

1408 

7 

616 

10 

880 

92 

25 

2300 

32 

2944 

24 

2208 

96 

32 

3072 

42 

4032 

32 

3072 

100 

19 

1900 

13 

1300 

23 

2300 

104 

4 

416 

4 

416 

11 

1144 

108 

3 

324 

1 

108 

I 

100 

9504 

100 

9500 

100 

9604 

Average 

95.04 

95.00 

96.04 

TABLE  63. 
Observations  on  the  Just  Perceptibi^e  Negative  Difference.     SUBJECT  VII. 


I 

III 

IV 

Tk 

N], 

l-k^'k 

Nk 

rkNk 

Nk 

l-kNk 

84 

2 

168 

88 

9 

792 

1 

88 

16 

1408 

92 

28 

2576 

24 

2208 

27 

2484 

96 

28 

2688 

31 

2976 

35 

3360 

100 

23 

2300 

31 

3100 

16 

1600 

104 

10 

1040 

5 

520 

2 

208 

108 

2 

216 

8 

864 

2 

216 

y; 

100 

9612 

100 

9756 

100 

9444 

Average 

96.12 

97.56 

94.44 

204 


PROBLEMS  OF  PSYCHOPHYSICS 


1— 1 

> 

it 

U 

00   00  00  Tti   o   cc   o 

CO    00    «    'M    O    CD    05 

.^    CI    C-1    O    O    1^    CI 

C-l    t-    »    t-    ■-'    -^ 

0) 

X 
X 
IN 

T* 

o 

CD 

05 

a 
-Z 

(M    to    »    ■*    O    t^    IM 
C-I    t-    05    t-    T^    •-< 

§ 

CO 

> 

00    •*    N    CO    O    CO    (M 

CD    O    lO    t^    O    t^   M 

.-H    O)    •*    -H    lO    05    ■<)< 

(M    t^    O    lO    ^ 

i 

X 
IN 

CO 
CO 

in 

05 

IM    CO    -H    CO    lO    05    ■* 

CO    M    O    lO    rt 

CO 

> 

J4 

CO    O    (N    ■*    O    O    O 
CO    CO    t^    IN    O    ■*    -"f 

CO   05  CO   CO   o   o   o 

(N 
t^ 

X 
IN 

■*   Lt.  CO   05   c   o   o 
■*   ,-<   CO   >o   .-1 

o> 

<N 

> 

2 

i 

COOOCO'^OCDO            Q 

COOOCOCOO'O'*            ?1 

CONOsOOO-*iO             ■* 

(N    05    00    lO    "-I                    00 

c 

o 

O) 

2 

Tj<    CO    CO    ■*    00    Til    10 
<M    O    00    <0    M 

g 

00    CD    •*    O    O    O    M 
CO   ^   00   TJ<   O   o   t^ 
•-1    CD    t~    00    O    CO    03 
■*  lO  ■*  cq 

o 

X 
C5 
X 
CO 

LO 

Oi 

it 
2 

r>\  t^  M  lo  o  lo  Oi 

lO    CD    ■*    C) 

o 

o 

K 

a. 

CO    t    X    IN    O    ■*    <M 

lO  (N  00  lo  o  o  ^^ 

t^    M     ■*     I^    C)     t^    05 

Tf    o    O    OC    (N 

CO 

§ 

X 
CO 

(N 

OS 

M 
2 

OS    00    ■*    IN    <N    CO    0> 
•*    rH    rt    00    (N 

i 

- 

00    (N    CD    p-1    O    ■*    •* 
O    CO    CO    U7    O    M    (M 
O    00    IN    r-    CO    CO    CO 
.-1    t~    (N    O    •* 

CO 

o 

CO 

CO 

CO 
05 

J4 
2 

IN    0!    CO    C-l    CO    CD    CO 
•^    00    CO    1^    il> 

X 
C5 
CO 

AS 

■*    X    IN    CO    O    ■*    00 
X    X    O!    05    O    O    O 

c^, 

ID 

§> 

> 

APPENDIX 


205 


TABLE  65. 

Differences  between-  Observed  and  Calculated  Values  of  the  Just  Perceptible 

Negative  Difference. 


Differences  between  observed  and  calculated  values. 

Subject 

Calculated  value 

IVa 

I 

III 

IV 

Totals 

I 

93.08 

-1.28 

-0.80 

+  1.35            +1.06 

+0.08 

II 

95.28 

+  0.68 

+  0.88 

0.00 

-1.72 

-0.04 

III 

97.40 

-0.88 

+  0.23 

+  0.44 

+  0.40 

+  0.05 

IV 

95.23 

-0.91 

-0.20 

+  0.57 

-0.18 

V 

94.20 

+  0.88 

+  1.68     1       -2.52 

+  0.02 

VT 

95.52 

-0.48 

-0.52            +0.52 

+  0.16 

VII 

96.04 

+  0.08 

+  1.52            -1.60 

0.00 

TABLE  66. 
Values  of  the  U's  for  the  Determination  of  the  Just  Imperceptible  Negative 

Difference. 


I 

II 

III 

IV 

V 

VI 

VII 

84 

0.0644 

0.0667 

0.0044 

0.0400 

0.0400 

0.0267 

0.0200 

88 

0.1351 

0.1286 

0.0442 

0.0736 

0.0704 

0.1331 

0.1012 

92 

0.3362 

0.2414 

0.1247 

0.2068 

0.2669 

0.2380 

0.2285 

96 

0.3085 

0.3104 

0.2480 

0.3806 

0.4421 

0.3071 

0.3533 

100 

0.1368 

0.1944 

0.3588 

0.2362 

0.1487 

0.2341 

0.2178 

104 

0.0188 

0.0528 

0.2067 

0.0596 

0.0309 

0.0569 

0.0745 

108 

0.0003 

0.0056 

0.0128 

0.0031 

0.0010 

0.0040 

0.0046 

I 

1.0001 

0.9999 

0.9996 

0.9999 

1.0000 

0.9999 

0.9999 

R 

0.0000 

0.0001 

0.0004 

0.0001 

0.0000 

0.0001 

0.0002 

206 


PROBLEMS  OF  PSYCHOPHYSICS 


> 

C    CO    O    X    C    O    00 
O    lO    C]    O   X   X    « 
X    O    CI    ^    1^    Tf    C3 

:d  o  o  05  i^  r~  •* 
T-^  X  ^  CO  ^  t^  d 

C^    CO    CI 

CI 

iq 

id 

C2 

> 

2.2428 
11.7128 
21.8960 
29.4816 
23.4094 
5.9176 
0.4320 

CI 
CI 

rs 
o 
id 

05 

> 

o  ci  X  to  in  to  o 

O    LO    Tfi    -H    X    CO    X 

CO  m  lo  •*  to  -H  o 
CO  >-;  o  •*  X  CI  '-; 
CO  CO  'i^  ci  Tr  CO  d 

Cl    ^   rt 

C3 

> 

O    =0    to    to    O    Tl"    X 
O    to    LO    t^    T)<    X    ■* 

S  r-  CI  CO  c;  03  CO 

CO  •*  p  lO  CD  .-;  CO 

CO  to  oi  CO  CO  CO  d 

^  CO  C) 

CI 

t-- 

UO 

id 

o 

a 

to   CO    -#    O    lO    X    ■* 
05    05    CI    X    t^    CO    CI 

CO  X  t~  o  r~  o  X 

CO    X    ■*    X    X    ^    CO 
d    CO    -^    CO    "-d    ^    rH 
rt    CI    CO    CI 

CO 
to 

05 
CJ 

X 

CI 

^~* 

X    X    X    •*    O    CI    X 
CI  CO  X  X  CI  -^  Tit 
O   -^    O    O    •*    35    O 
CO    CO    C)    t-;    •*    •*    CD 

i-d  '^  ci  d  d  id  d 

^    Cl    Cl    -> 

X 

Tf 

to 

■* 
••ji 

C5 

HH 

CD    X    ■*    O    lO    CI    ■* 
03   X    O    CO    CO    "O    CI 
O   X    CO   -"I    t^    "-O    CO 

^  X  CI  CO  CD  c>  q 
id  -^  d  d  CO  --t  o 
^  CO  ci  rt 

X 

o 

lO 

CO 

C5 

^ 
U 

T)<   X    IM   CO    C    *^   X 

X  X  o  o  o  o  o 

^ 

3    X    O    X    C    C    TJ< 

o 

C    CI    •*    CI    O    C)    Tl< 

o 

CI    o    CI    —    o    O    lO 

05 

■    ■ 

rt  CD  p  q  X  r-;  CO 

p 

1— 1 

rH  CO  ■*  CO  t-  id  CO 

ci 

> 

■*    X    CO    LO    1^    O    lO 

lO 

■-1    W    O    CI    -H    X 

-H  CO  CI 

OS 

CI    ■*    o   CD    O    ■*    O 

CD 

lO    CO    CI    CO    O    C    CO 

CO 

ci  CI  CO  CO  ■<»'  CO  LO 

,_, 

CO    t-    ■*    CI    05    ■*    CD 

00 

> 

x  d  -^  d  d  id  CO 

cd 

X  CO  ^  CO  •*  -1  •* 

CO 

1-1    O    O    X    CO    CD 

o 

rt    CI    CI    CI 

o 

O    CO   CO   CO   O    •*    o 

CI 

O   t-  -H   CO   o  ■^   ■* 

■^  r^  -^  O  i^  '-'  CO 

X 

> 

CI    -H    p    CO    X    CI    CO 

lo 

CI  id  d  •*'td  Ti<  ,-< 

cd 

X    -^t    LO    t^   X   CO    " 
CI    lO    CI    O    "*    CO 

a> 

O) 

CI    ■*    .-1 

X 

O    ■*    CI    to    O   CD    ■* 

C5 

O    X    LO    05    O   CO    X 

IC 

'i'    LO    LO    O    O   CO    LO 

LO 

> 

CI    p    CO   CO    ■^.    CD    ^ 

CO 

ci  d  d  r-^  CI  tiJ  cd 

cd 

X    CD    LO    O    CD    Tt"    CO 

lO 

CI    lO    t^   LO    CO    CD 

.-<    CO    N 

Cs 

■«<    X    X    O    O    CI    CI 

Ttl 

CO  ■<)<  o  X  o  r^  o 

CO 

•<9<    X   CO    CO    lO    CD    Ol 

t^ 

HH 

p  CI  T)<  in  r--  p  CI 

o 

H 

^  ci  id  LO  t~^  LO  d 

1^ 

CO   •*    lO   X    X    CO   •* 

X 

CO    C    CI    lO    CI    rt 

to 

T-1  CI  CO  CI 

03 

CI    ■*    CO    •*    O   X    •* 

X 

LO   X    O   CO    O    ■*    X 

CI 

CO    t~    O    ■*    O    X    -H 

t^ 

t—1 

p  00  CI  p  CI  p  CO 

05 

^~* 

d  LO  cd  d  •*  — <  id 

d 

t^  o  ^  to  •'1'  r-  CO 

in 

Tj(    O    O    X   05    LO 

cs 

CI    CI    •-< 

X 

■*   ■*    X    O    O   X    CI 

CO 

CO    ■*    CD    CO   O    O    Oi 

CO 

O  -^  C5  CO  lO  •*   o: 

•* 

•*  CI  LO  -^  CO  CO  •* 

00 

*~* 

■*'  cd  Ld  cd  t-^  cd  cd 

cd 

LO    'I'    •*    ■*    CD    O 

CO 

■*  O  X  X  CO  CI 

r^ 

r^     CI     CI     .-1 

X 

^ 

■*  X   CI  CO  c  •*   X 

^ 

I. 

X   X   05  C3  o  o  o 

APPENDIX 


207 


TABLE  69. 
Observations  on  the  Just  Imperceptible  Negative  Difference.     SUBJECT  1. 


IVa 

I 

III 

IV 

Tk 

N.        • 

TkNfe 

Nk 

'    r^N^ 

Nk 

i      l-kNk 

Nk 

J-kNk 

84 

7 

588 

6 

504  * 

7 

!           588 

5 

420 

88 

12 

1056 

13 

1        1144 

9 

1            792 

8 

704 

92 

33 

3036 

50 

4600 

33 

3036 

18 

1656 

96 

39     1 

3744 

18 

1728 

36 

3456 

40 

3840 

100 

9 

900 

12 

1200 

13 

1300 

23 

2300 

104 

1 

104 

2 

208 

6 

624 

V 

100 

9324' 

100 

9280 

100 

9380 

100 

9544 

Average 

1 

93.24 

92.80 

j       93.80 

95.44 

TABLE  70. 
Observations  on  the  Just  Imperceptible  Negative  Difference.     SUBJECT  II. 


IVa 

- 

I 

III 

IV 

Tk 

Nk 

I-k  Nk 

Nk 

TkNk 

Nk 

'     i-^Nk 

Nk 

1    r^Nj, 

84 

5 

420 

8 

672 

3 

252 

9 

756 

88 

12 

1056 

14 

1232 

13 

1144 

12 

1056 

92 

19 

1748 

23 

1        2116 

21 

1932 

28 

2576 

96 

35 

3360 

26 

j        2496 

38 

3648 

31 

2976 

100 

19 

1900 

22 

2200 

15 

1500 

15 

1.500 

104 

9 

936 

5 

520 

10 

1040 

4 

416 

108 

1 

108 

2 

216 

1 

108 

I 

100 

9528 

100 

9452 

100 

[         9516 

100 

9388 

Average 

95.28 

1      ^^••^^' 

j        95.16 

1      93.88 

208 


PROBLEMS  OF  PSYCHOPHYSICS 


TABLE  71. 
Observations  on  the  Just  Imperceptible  Negative  Difference.     SUBJECT  III. 


IVa 

I 

[II 

IV 

Tk 

Nk 

TkNk 

Nk 

TkNk 

Nk 

TkNk 

Nk 

J-kNk 

84 

2 

168 

_ 

88 

8 

704 

5 

440 

92 

14 

1288 

6 

552 

3 

276 

11 

1012 

96 

24 

2304 

21 

2016 

31 

2976 

41 

3936 

100 

33 

3300 

47 

4700 

61 

6100 

36 

3600 

104 

18 

1872 

20 

2080 

2 

208 

10 

1040 

108 

1 

108 

1 

108 

3 

324 

2 

216 

I 

100 

9744 

100 

9896 

100 

9884 

100 

9804 

Average 

97.44 

98.96 

98.84 

98.04 

TABLE  72. 
Observations  on  the  Just  Imperceptible  Negative  Difference.     SUBJECT  IV. 


I 

III 

£W 

I-k 

Nk 

TkNk 

Nk 

TkNk 

Nk 

TkNk 

84 

1 

84 

4 

336 

5 

420 

88 

9 

792 

7 

616 

1 

88 

92 

15 

1380 

31 

2852 

11 

1012 

96 

46 

4416 

36 

3456 

39 

3744 

100 

21 

2100 

17 

1700 

28 

2800 

104 

8 

832 

5 

520 

14 

1456 

108 

2 

216 

I 

100 

9604 

100 

9480 

100 

9736 

Average 

96.04 

94.80 

97.36 

APPENDIX 


209 


TABLE  73. 
Observations  on  the  Just  Imperceptible  Negative  Diffekence.     SUBJECT  V. 


I 

III 

IV 

fk 

Nk 

TkNk 

Nk 

TfcNk 

Nk 

Tk   Nk 

84 

2 

168 

2 

168 

7 

588 

88 

12 

1056 

8 

264 

8 

704 

92 

22 

2024 

21 

1932 

31 

2852 

96 

38 

3648 

44 

4224 

5.0 

4800 

100 

22 

2200 

23 

2300 

4 

400 

104 

4 

416 

7 

728 

108 

y 

100 

9512 

100 

9616 

100 

9344 

Average 

95.12 

96.16 

93.44 

TABLE  74. 
Observations  on  the  Just  Imperceptible  Negative  Difference.     SUBJECT  VI. 


- 

I 

III 

IV 

Tk 

Nk 

rkNfc 

Nk 

r^Nk 

Nk 

TkNk 

84 

7 

588 

88 

16 

1408 

13 

1144 

9 

792 

92 

36 

3312 

16 

1472 

25 

2300 

96 

19 

1S24 

39 

3744 

35 

3360 

100 

18 

1800 

29 

2900 

23 

2300 

104 

4 

416 

3 

312 

6 

624 

108 

2 

216 

I 

100 

93.48 

100 

9572 

100 

9592 

Average 

93.48 

95.72          1 

95.92 

210 


PROBLEMS  OF  PSYCHOPHYSICS 


> 

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t-    CO    00    -^    CO    00 

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C) 

CO 
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Oi 
Cl 

1 

OS 

a, 
2 

GO    W    CO    -^    CO    OC 
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1 

o 

?3 

> 

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2 

00    •^    ■*    00    O    (M    <£> 

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CI 

o 

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t^    CO    t^    CO    O    CO    IM 

CO  t^  o  t-  »-i 

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CO 

> 

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CI 

00 

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2 

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o 
o 
CO 

> 

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00    ■*    (M    CO    CD    00    CJ 

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00 

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c» 

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CD 

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1 

00 
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CO 

CO 

Cl 

CO 

00 

o 

2 

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'H  CO  -H  r-  in 

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2        ooot-OjO'Hro 

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00 

00 
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05 

2 

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o 
o 

l-l 

2 

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--I  CD  CO  r^  r^  ci 

Cl    CO    Cl    Cl    lO 

00 
Cl 

LO 

en 

oc 

CO 

o 

2 

lO    O    -^    CO    t^    03 
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o 
o 

1 

^    I 
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: 

rt<   X    Cl    CD   C    ■*   CO 
00    QC    O    35    C    C    C      1 

1 
1 

M 

> 
< 

APPKN'DIX 


211 


TABLE  75. 
Observations  on  the  Just  Imperceptible  Negative  Difference.     TABLE  VII. 


I 

III 

IV 

Tk 

Nk 

rfcNk 

Nk 

1 

TkNk 

Nk 

i-kNk 

84 

88 

2 

176 

6 

528 

92 

12 

1104 

16 

1472 

19 

1748 

96 

37 

3552 

32 

3072 

44 

4224 

100 

32 

3200 

42 

4200 

27 

2700 

104 

13 

1352 

7 

728 

3 

312 

108 

4 

432 

3 

324 

1 

108 

I 

100 

9816 

100 

! 

9796 

100 

9620 

Average 


98.16 


97.96 


96.20 


Table  76  will  be  fou.n'd  on  the  preceding  page 


TABLE  77. 
Differences  between-  Observed  and  Calculated  Values  op  the  Just  Imperceptibls 

Negative  Difference. 


Difference  between  observed  and  calculated  values. 

Subject 

Calculated  Value           IVa                  I 

III 

IV 

Totals 

I 

1 
93.51            j        -0.27             -0.71            +0.29            +1.93 

+  0.31 

II 

94.46 

+  0.82             +0.06      '       +0.70             -0.58 

+  0.25 

III 

98.30 

-0.86             +0.66             +0.54             -0.26 

+  0.02 

IV 

95.56 

+  0.48      !        -0.76            +1.80 

+  0.51 

V 

94.74 

+  0.38     1       +1.42 

-1.30 

+  0.17 

IV 

95.09 

-1.61 

+0.63 

+0.83 

+0.05 

VII 

95.55 

j 

+  2.61 

+  2.41 

+  0.65 

+  1.89 

212 


PROBLEMS   OF    PSYCHOPHYSICS 


E 


Ji    C 


eti 

XI 

c 

1 

S 

« 

05   O   '— '   o 
CD    O    O    CO 


(U 

CI    I-    CI    o 

(D 

o  o  c  -^ 

!C 

1   1  +  + 

P 

lO    lO    lO   lO 

c 

00    O    ■*    lO 

£ 

IM    00    CO    O 

-H      O     ^     ^ 

it! 

1    1  +  + 

p 

o  o  o  o 

Tl"    O    O   ■<*' 


05    05    Oi    OS 


o  o  o  o 

O    00    C5   ■* 


Oi    05    05    OS 


-H  «D  "O  r^ 

CI     O    <M     t--; 

dodo 


o  o  o  o 

CI    O    00   « 
O    CI    00   03 


«    O    CJ    CI 

-^  d  d  d 

+  +  +  + 


o  o  o  o 


^ 

f>i 

(-) 

o 

•o 

CO 

LJ 

o 

> 

,_, 

^ 

O 

(^ 

O 

o 

O 

o 

!^ 

•"• 

^^ 

»— < 

^ 

O 

OJ 

t-H 

H- < 

> 

u 

> 

^ 

I    jDaCqns 


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lO 

lO 

lO 

lO 

ra 

C/) 

X 

CD 

•o 

o 

o 

O 

o 

+ 

1 

+ 

1 

o  o  o  o 
oc  CI  CO  00 

CI    lO    -H    00 


CI  CI  CI  00 

00  00  O  -H 

CD  rH  o  I- 

o  d  d  -H 

+  +  +  I 


o  o  o  o 

CD  CD  CC  « 

oi  ^  CI  in 

lO  d  fD  03 

Ci  a  a  Oi 


CO  ■*   ■*   ■* 

05    ■*    00    00 


rt  o 

C) 

■* 

o  o 

T-H 

r-H 

+  1 

1 

1 

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o 

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CD   CI 

fTi 

00 

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o 

00 

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00 

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o  o 

05 

8 

O 

289 
815 
281 

rt 

o  o  o 

1 

+ 

1  +  + 

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§ 

a 

> 

CO 

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CI    C)    00 

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§ 

00    05    00 
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> 

'-'!=;> 

II 

losfqns 

CD 

■rf 

TjH  CO 

CO 

CO 

CO 

LO      CI 

o 

c 

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1 

+ 

4-    1 

o 

c 

o  c 

CO 

■*      Tf 

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C: 

CO   o 

t^ 

rT' 

00:      00 

o 

C3;  05 

f  CD    CO  CD 

00  CO    CO  05 

CO  CI    T)<  CO 

d  d  d  d 

I  +  +  + 


o  o  o  o 

CI  'I'  ■*  o 

LO    CO    OC    00 


G:    Ci    Oi    Ci 


<N  00  00  00 

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t^  c  to  CO 

d  -H  ^  --^ 

i  +  +  + 


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ca 

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> 

h— .  *— ' 

III  issfqi^s 


APPENDIX 


21.3 


CO 

t- 

«  1 

00 

o 

c 

■«< 

t^ 

00 

o 

o 

^^ 

i    + 

1 

+ 

o 

o 

1  ■* 

o 

1   o 

00 

CO 

<£> 

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r^ 

1   O 

m 

» 

I  + 


P  Q 


o 

o 

o 

c^ 

CO 

o 

CO 

o 

00 

•* 

in 

>o 

05 

o 

m 

o   S2 

a  i5 


r^ 

t^  00 

(D 

O  C-l 

O 

c-1  CO 

d 

1 

c  o 
1  1 

o 

O  05 

OS  00 

-   00 

r^  r^ 

Oi 

Ol  05 

re 

o 

50 

r- 

CO 

IN 

o 

o 

^H 

1  + 

1 

+ 

o 

n 

O   1 

00 

(D 

CD 

■» 

00 

t^ 

o 

1  o 

o 

OS 

1 

> 

1 

00 

CO 

2 

■* 

o 

CO 

o 

^^ 

»M 

+ 

+ 

1 

CO  CO  1^ 

00  00  .-I 

00  to  >c 

d  .-<  <m" 

+  +    I 


00 

o 

00 
00 

ffl 

lO 

in 

.-1 

+  I   1 


<     s 


to 

■* 

■M 

00 

CO 

CO 

CO 

M 

o 

C  1 

1   + 

+ 

1 

o 

Tf 

M 

CO 

M 

05 

r^ 

CO 

o 

05 

1 

„ 

>  ' 

A I   loafqns 


A   l^sfqns 


M 

X 

X 

•M 

C^ 

CO 

X 

— 1 

o 

d 

1 

+ 

+ 

O 

n 

X 

rl 

M 

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l'- 

a> 

CO 

lO 

o 

o 

05 

03 

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1 

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o  o 

Tft 

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m 

IC  CD 

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03  03 

X  X  (N 

t^  CO  IN 

IN  X  CO 

rt  d  ^ 

1   I  + 


IN    CD    X 
03    03    '-H 


a> 

M 

C^l 

^" 

n 

C 

o 

^H 

■N 

CO   1 

' 

+ 

+ 

03 

O 

o 

X 

M 

M 

t^ 

,— ( 

M    1 

1    "* 

O 

O   ^ 

1 

„ 

> 

M    CI 
+    + 


IN  (M 

X 

03  cq 

ffl 

O  UJ 

•fl 

O  -H 

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1 

o  o 

O 

IN  CO 

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Tt" 

CO  t^ 

Tjt 

03  o; 

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o  ->  o 

X    O    CD 
CO    O    CO 


CO 

IN 

CO 

CD 

03 

o 

CO 

IN 

IN 

^H 

+ 

+ 

+ 

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IN 

C 

03 

CO 

t^ 

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(-; 

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o 

o 

> 

lA  psfqng 


II A  533.-q"s 


214 


PROBLEMS  OF  PSYCH(^PH YSICS 


TABLE  79. 
Deviations  of  Observed  Values  from  the  Theoretical  Results  for  the  Thresholds. 


Threshold  in  direction  of  increase. 

Threshold  in  direction  of  decrease. 

Theoretical  value     99.452 

Theoretical  value     93.297 

,_, 

Series 

Observed  value 

Difference 

Observed  value 

Difference 

o 

3 

IVa 

I 

III 

IV 

100.020 
99.760 
99.440 
99.180 

+  0.568 
+  0.308 
-0.012 
-0.272 

92.520 
92.540 
94.115 
94.770 

-0.777 
-0.757 
+  0.818 
+  1.473 

Average 

0.290 

Average              0.956 

Theoretical  value 

98.835 

Theoretical  value 

94.871 

f— < 
U 

3 

IVa 

I 

III 

IV 

99.437 
98.670 
98.600 
98.233 

+  0.602 
-0.165 
-0.235 
-0.652 

95.620 
95.340 
95.220 
93.720 

+  0.749 
+  0.469 
+  0.349 
-2.151 

Average 

0.414 

Average 

0.929 

Theoretical  value 

99.284 

Theoretical  value 

97.850 

o 

3 

IVa 

I 

III 

IV 

98.560 

98.340 

101.020 

100.420 

-0.724 
-0.944 
+ 1.736 
+  1.144 

96.980          1 
98.300          1 
98.340          1 
97.920 

-0.870 
+  0.550 
+  0.490 
+  0.070 

Average 

1.137 

Average 

0.495 

Theoretial  value     98.083 


I 
III 
IV 


98.400 
97.780 
98.550 


+  0.317 
-0.303 
+  0.467 


Average 


0.362 


Theoretical  value     95.393 


95.180 
94.915 
96.580 


-0.213 
-0.478 

+  1.187 


Average 


0.626 


Theoretical  value 

97 

.142 

Theoretical  value 

94.469 

> 

o 

'2 

3 

I 
III 

IV 

'            99.780 
97.300 
94.960 

+  2.638 
+  0.158 
-2.182 

95.100 
96.020 
92.560 

+  o.63i 
+ 1.551 
-1.909 

Average 

1.659 

Average 

1.364 

Theoretical  value 

99.863 

Theoretical  value 

95.307 

> 

I 
III 
IV 

98.530 

100.460 

j           102.025 

I 

-1.333 
+  0.597 
+  2.162 

94.260 
95.360         1 
95.980 

-1.147 

+  0.053 
+  0.673 

Average 

1.364 

Average 

0.624 

Theoretical  value     99.859 


Theoretical  value     95.793 


I 
III 
IV 


100.180 
99.426 
99.240 

Average 


+  0.321 
-0.433 
-0.619 

0.458 


97.140 
97.760 
95.320 


Average 


+  1.347 
+  1.967 
-0.473 

1.262 


API'KNDIX 


215 


TABLE  SO, 

THEOKICTK'AL    VaI.I'I'.S    of    IHB    Thuhsiiui.us. 


Subject 

Threshold  in  direc- 

Threshold in  ilirec- 

tion  of  increase         , 

lion  of  decrease 

I 

99.45 

93.30 

II 

98.  S3 

94.87 

in 

99.28 

97.85 

IV 

98.08 

95.39 

V 

97.14 

-  94.47 

IV 

99.86                   i 

95.31 

vn 

99.86                   ' 

95.79 

Interval  of 
uncertainty 

6.15 
3.96 
1.43 
2.69 
2.67 
4.56 
407 


TAI3LE  SI. 
Observed  V.\i.l'es  of  the  Thresholds. 


Subiect 

Threshold  in  direc-       } 

Threshold  in  direc- 

Interval of 

tion  of  increase 

tion  of  decrease 

uncertainty 

I 

99.60 

93.49 

(ill 

II 

98.71 

94.98 

3.74 

III 

99.58 

97.88 

1.70 

IV 

98.24 

95.56 

,      2.68 

y 

97.34 

94.57 

2.78 

IV 

100.33 

95.20 

5.12 

VII 

99.62 

96.74 

2.88 

TABLE  82. 

R.\TIOS   OF   THE   XU.MBERS   OF    "  HR.\  VI  liK-GU  ESS  "     AND    "  LIGHTER  GUESS  "-JUDGMENTS. 


84 
88 
92 
96 
100 
104 
108 


I 

II 

TV 

V 

VI 

VII 

0.3333 

0.6667 

0.2500 

0.3333 

1.0000 

0.0000 

0.2727 

0.8214 

1..5000 

2.7.500 

0.5455 

0.4500 

0.6021 

1.5758 

0.5909 

1.2105 

1.7500 

0.6571 

1.0101 

1.2745 

0.8485 

4.9000 

0.S302 

0.9649 

3.8605 

1.3478 

1.0769 

5.1429 

1.3830 

1.4054 

2.7273 

1.5000 

3.0000 

28.0000 

3.0714 

2.6667 

3.8000 

1.2500 

4.0000 

10.0000 

2.2222 

4.2500 

TABLE  83. 
Relative  FREguENCiEs  of  "G"  Judgments. 


Comparison 

I 

n 

III 

Weight 

84 

0.0622 

0.0444 

0.0044 

88 

0.1244 

0.1133 

0.0200 

92 

0.3311 

0.1889 

0.0600 

96 

0.4422 

0.2578 

0,0778 

100 

0.4644 

0.2400 

0.1 489 

104 

0.0911 

0.0889 

0.0467 

108 

0.0533 

0.0800 

0.0156 

IV 

V 

VI 

VII 

0.0167 

0.0133 

0,0200 

0.0133 

0.0.500 

0.0500 

0,1133 

0.0967 

0,1167 

0.1400 

0.2200 

0.1933 

0,2033 

0  1967 

0.3233 

0.3733 

0,1  SOO 

0.1433 

0,3733 

0.2967 

0,0667 

0.0967 

0.1900 

0.1833 

0,0500 

0.0367 

0.0967 

0.1400 

216 


PROBLEMS  OF  PSYCHOPHYSICS 


> 
> 

U 

«0  Cl  to  M  O  O  CO 

«  >0  «  O  O  IM  CO 

c<3  >n  «  t-  ra  r^  10 

(M  IC  O  00  lO  'J' 

CO 

00 

CO 

CO 

o 

00 
05 

•*  Oi  X  (M  m  lO  M 

iM  ira  — '  CO  lO  •* 

0-. 

00 

CO 

■*  <M  Cl  C-l  O  CO  CI 

O  0>  t-  r-i  o  CI  CC 

O  05  O  M  CI  0>  — ■ 

CI  ;D  05  '— '  lO  CO 

o 

05 

CO 

>o 

CO 
05 

CO  ■*  CO  t-  CI  t-  a 

CO  CO  c;  -H  ic  CI 

"S 

> 

it 

2 

CO  O  •*  ■*  O  CO  00 

CO  CI  CD  CO  C  -I  x 

CO  CO  *  CO  re  C;  — 
— '  CO  L-:  •*  CO  ^ 

00 

00 
CO 

OS 

00 
o> 

CD 
OS 

•*  lO  CI  O!  CO  CT:  -1 

i-H  •^  IC  ■*  CI  — 1 

CO 

> 

l-i 

J! 

O  O  O  CO  o  o  o 
CI  CI  CI  uO  O  CC  CI 
■*  CO  CI  00  ■*  O  CO 

-H  CO  -o  .o  CI  -. 

2 

05 

^ 

2 

lO  lO  >0  — 1  ■*  1.0  o 

-H  CO  CO  LO  CI  -H 

CI 

00 

CI 

CD 

i 

t-H 

2 

on  <M  -^  o  o  00  CO 

CO  05  00  CO  O  CO  iCi 
^  1-  t  CO  1-  -1  t- 

CI  CO  CO  CI 

00 

o> 

M 
2 

ca  05  t^  lo  r-  -H  t^ 

(N  CO  CO  M 

00 
CD 

M 

AS 

2 

O  00  O  CO  O  O  tn 
00  CO  C)  CO  O  CO  CO 

CO  ■*  00  -^  00  r^  Of) 
-H  ■ijl  t^  -H  O  •*  CO 

CI 

05 

CO 

CO 

■<»■ 

CO 
05 

O  -H  lO  CO  00  O  CO 

CI  O  00  -^  O  ■*  CO 

CO 

CI  00  00  -"J"  O  ■*  CI 
lO  CI  O  O  O  CO  Ol 
CO  OS  1^  -^  03  CI  lO 
CI  Tf  CO  05  O  Ttl  W 

00 

CO 

CO 

o 

CO 

o> 

00  CO  C5  OS  05  -H  ■* 

N  lO  ■*  OS  O  T(i  c^ 
^  rt  M 

1- 

■*  SS  «  =  °  o  0 

30  so  05  M  vO   «   „ 

N 

t 

< 

.\PPElSrDIX 


217 


J 

O  CO  «  o  o  o  o 

;0  C^l  CO  Tf  ^^  lO  CI 

05  05  t-  -^  M  q  c 
d  d  d  d  d  d  d 

> 

X 

CO  r~  t^  r^  O  CO  o 

CO  CO  !D  CO  O  CO  O 
M  C-l  --  lO  r-^  CO  CO 

O  O  --I  CO  CO  00  o> 

d  d  d  d  d  d  d 

o 

t^  O  t^  CO  o  t^  o 

CO  O  CO  CO  O  CO  o 

Si  ira  !i<  o  M  CO  in 
o  q  r^  N  -^  o  o 
o  o  o  o  o  o  o 

J 

CO  CO  OS  O  O  O  r^ 
lO  lO  00  O  O  O  CO 

O!  lO  CD  O  00  CO  C"l 

OS  o  00  r-;  CO  q  q 

d  d  d  d  d  d  d 

X 

0.0000 
0.0244 
0.0711 
0.2222 
0.4711 
0.8933 
0.9578 

■*  o  o  00  O)  r^  CO 

■^  O  O  t-  00  CO  "O 
O  C-1  CD  1-  ^  ■*  -^ 

o  w  o  q  ^  c  q 
d  d  d  d  d  d  d 

.1 

1 

►J 

CO  C-)  O  35  —1  CO  CO 
CO  M  O  »  — '  »0  lO 

CO  CO  O  T<  CO  OS  -^ 

05  X  t»  •»  M  q  q 

d  d  d  d  d  d  ci 

X 

o 

M  Tf  -H  CO  05  CD  -^ 

CI  'J'  -H  CO  «  "O  •<i< 

CI  (M  — 1  o:  M  — •  O 

o  o  -<  cj  ic  00  q 
d  d  d  d  d  d  d 

Tji  CO  05  M  o  a>  c 

•*  CO  X  t-  O  X  o 
Tl<  -^  »  lO  "1"  X  X 

C  rH  ^  CI  CI  C  O 

d  d  d  o  d  o  d 

Hil 

J 

CO  CO  O  CO  c^  CO  r- 
lO  in  O  *0  CI  CO  CO 
CO  lO  00  CO  d  '^  o 
0>  00  lO  CO  ^  o  o 

d  d  d  d  d  d  d 

X 

o 

1 

CI  O  05  Cl  CO  CO  o 
CI  O  X  CI  CO  iC  c 
O  C-I  X  CI  --  0-.  ■* 
O  O  O  CI  ■•I'  X  05 

d  d  d  d  d  d  d 

0.0622 
0.1244 
0.3311 
0.4422 
0.4644 
0.0911 
0.053;; 

i 

1 

■<r  X  CI  CO  o  V  X 
W  X  0>  05  o  o  o 

hJ 

Q  t^  o  t^  t^  o  o 

O  CO  O  CO  CO  o  o 
X  05  •*  iC  CO  CD  ■<>< 
05  X  t^  •0'  C\|  o  o 

d  d  d  d  d  d  d 

i_i 

X 

r~  i~  r^  o  h-  t^  s 

CO  CO  CO  O  CO  CO  o 

o  o  CO  w  CO  o  CI 

O  O  O  -H  ■*  I-  X 

> 

o  o  o  o  o  o  o 

o 

CO  r-  CO  CO  t-  CO  o 

CO  CD  CO  CO  CO  CO  o 
-1  05  O)  I-  O)  X  ■v 
C  O  ^  CO  CI  "  rt 

d  d  d  d  d  d  d 

J 

CO  CO  t^  o  t-  t^  o 
CO  CO  CO  O  CO  CO  o 
I,  CD  — >  o:  O  CO  CI 
OS  X  I^  TT  CI  o  o 

o  o  o  o  o  o  o 

> 

X 

i^  CO  CO  t^  c  CO  CO 

CO  CO  CO  CO  c;  CO  CO 

O  CI  CD  X  CI  -n-  X 
O  O  O  •-<  ■*  1^  00 

d  d  d  d  d  ^  d 

o 

O  CO  O  CO  CO  o  t^ 
O  CO  O  CO  CO  O  CO 
CI  «  CI  Cl  t'  05  05 

o  «  CI  CO  CO  -H  q 
d  d  d  d  d  d  d 

J 

o  r^  o  o  t^  CO  o 

O  CO  O  O  CD  CO  C 
CO  N  O  05  I-  CO  CI 

05  05  I-  CI  -■  q  q 
d  d  d  d  d  d  d 

^ 

1 

> 

X 

t-  M  c  CO  c;  o  w 

CO  CO  O  CO  C  O  CO 
d  CI  CO  -<  X  1-  -TJ" 

o  o  -^  "O  q  00  05 
d  d  d  d  d  d  d 

1 

o 

CO  C  C  I-  CO  t^  h; 

CO  O  O  CD  CO  CD  CO 

--  in  Tj>  05  -a'  C5  CO 
o  o  -H  ^  -.  q  q 

d  d  d  d  d  d  o 

T)i    X   M    CO   O    ■*    X 
X  X  05  05  o  o  c 


218 


PROBLEMS  OF  PSYCHOPHYSICS 


TABLE  86. 
Values  of  the  Psychometric  Fu.vctioms.     SUBJECT  L 

G  H  L 


84        1 

0.0622 

0.0022 

0.9356 

85 

-0.0167 

0.0433 

0.9734 

86 

-0.0039 

0.0434 

0.9605 

87 

0.0532 

0.0302 

0.9166 

88 

0.1244 

0.0200 

0.8556 

89 

0.1926 

0.0205 

0.7869 

90 

0.2501 

0.0337 

0.7162 

91 

0.2955 

0.0577 

0.6468 

92 

0.3311 

0.0889 

0.5800 

93 

0.3608 

0.1232 

0.5160 

94 

0.3881 

0.1576 

0.4343 

95 

0.4152 

0.1905 

0.3943 

96 

0.4422 

0.2222 

0.3356 

97 

0.4664 

0.2554 

0.2782 

98 

0.4827 

0.3020 

0.2153 

99 

0.4846 

0.3453 

0.1701 

100 

0.4644 

04133 

0.1223 

101 

0.4157 

0.5033 

0.0810 

102 

0.3347 

0.6168 

0.0485 

103 

0.2232 

0.7513 

0.0255 

104 

0.0911 

0.8956 

0.0133 

105 

-0.0398 

1.0294 

0.0104 

106 

-0.1328 

1.1186 

0.0142 

107 

-0.1284 

1.1125 

0.0159 

108 

0  0533 

0.9400 

000(i7 

T.\[ 

BLE  87. 

Valu 

ES    OF    THE    PSVCHOME 

TRic  Functions.     S 

J EJECT   IL 

"^    Tk 

G 

H 

L 

84 

0.0444 

0.0222 

0.9334 

85 

0.0614 

0.0285 

0.9101 

86 

0.0786 

0.0258 

0.8956 

87 

0.0959 

0.0227 

0.8814 

88 

0.1133 

0.0244 

0.8623 

89 

0.1310 

0.0337 

0.8353 

90 

0.1497 

0.0516 

0.7987 

91 

0.1690 

0.0777 

0.7533 

92 

0.1889 

o.iiu 

0.7000 

93 

0.2088 

0.1505 

0.6407 

94 

0.2279 

0.1946 

0.5775 

95 

0.2447 

0.2424 

•    0.5129 

96 

0.257S 

0.2933 

0.4489 

97 

0.2654 

0.3471 

0.3875 

98 

0.2650 

0.4040 

0.3301 

99 

0.257S 

0.4643 

0.2779 

100 

0.2400 

0.5289 

0.2311 

101 

0.2125 

0.5975 

0.1900 

11)2 

0.1761 

0.6698 

0.1541 

U)3 

0.1334 

0.7436 

0.1230 

104 

0.0889 

0.8156 

0.0955 

105 

0.0495 

0.8795 

0.0710 

106 

0.0251 

0.9255 

0.0494 

107 

0.0294 

0.9403 

0.0303 

108 

0.0,800 

0.9044 

0.0156 

APPENDIX 


219 


TABLE  88. 
Values  of  the   PsvcHo.\rETRic  Function's. 


rk 
84 

K.i 
86 

sr 
ss 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

103 

104 

10.3 

106 

107 

108 


SUBJECT  III. 


0.0044 

0.0000 

0.9956 

-  0.0398 

0.0359 

1.0038 

-0.0347 

0.0403 

0.9944 

-  0.0080 

0.0325 

0.9764 

0.0201) 

0.0244 

0.9556 

0.043.1 

,  0.0226 

0.9339 

0.0.').")i> 

0.0299 

0.9142 

0.0612 

O.0464 

0.8924 

0.0600 

0.0711 

0.8689 

0.0577 

0.1022      ' 

0.8401 

0.0590 

0.1381 

0.8029 

0.0647 

0.1781 

'   0.7573 

0.0778 

0.2222 

0.7000 

0.0960 

0.2714 

0.6326 

0.1170 

0.3278 

0.5552 

0.1364 

0.3936 

0.4700 

0.1480 

0.4711 

0.3800 

0.1490 

0.5619 

0.2891 

0.1325 

066.55 

0.2020 

0.0975 

0.7786 

0.1239 

0.0667 

0.8933 

0.0400 

-0.0108 

0.99.55 

0.0153 

-0.0571 

1.0636 

-  0.0065 

-0.0629 

1.0657 

-  0.0028 

0.01.56 

0.9578 

0.0266 

TABLE  89. 
Values  of  the  Psycho.metric  Function'? 


fk 

G 

H 

84 

0.0167 

0.0233 

85 

0.0386 

0.0.532 

86 

0.04.53 

0.0.502 

87 

0.0469 

0.0370 

88 

0.0.500 

0.0267 

89 

0.0.581 

0.0276 

90 

0.0725 

0.0430 

91 

0.0926 

0.0736 

92 

0.1167 

0.1167- 

93 

0.1425 

0.1697 

94 

0.1672 

0.2-294 

95 

0.1883 

0.2925 

96 

0.20.33 

0.3.567 

97 

0.2106 

0.4205 

98 

0.2092 

0.48.39 

99 

0.1986 

0.5464 

100 

0.1800 

0.6100 

101 

0.1548 

0.67.56 

102 

0.1253 

0.7439 

103 

0.0948 

0.8142 

104 

0.0667 

0.8,S33 

105 

0()44,S 

0.9446 

106 

0.032S 

0.9863 

107 

0.03.37 

0.9904 

108 

0.0500 

0.9300 

SUBJECT  IV. 
L 


0.9600 
0.9082 
0.9045 
0.9161 
0.9233 
0.9143 
0.8845 
0.8338 
0.7666 
0.6878 
0.6034 
0.5192 
0.4400 
0.3689 
0.3069 
0.2.5.50 
().21(MI 
0. 1696 
0.1308 
0.0910 
0.0500 
0.0106 
-0.0191 
-0.0241 
0.0200 


220 


PROBLEMS    OF    PSYf'HOPHYSICS 


TABLE  90. 
Values  of  the  Psychometric  Functions.     SUBJECT  V. 


G 

H 

L 

84 

0.0133 

0.0267 

0.9600 

85 

0.0290 

0.1116 

0.8594 

88 

0.034S 

0.1031 

0.8621 

87 

0.0401 

0.0610 

0.8989 

88 

0.0500 

0.0233 

0.9267 

89 

0.0663 

0.0108 

0.9229 

90 

0.0883 

0.0313 

0.8804 

91 

0.1138 

0.0833 

0.8029 

92 

0.1400 

0.1600 

0.7000 

93 

0.1637 

0.2514 

0.5849 

94 

0.1822 

0.3465 

0.4713 

9.5 

0.1935 

0.4360 

0  3705 

96 

0.1967 

0..-)133 

0.2900 

97 

0.1917 

0.5744 

0.2339 

98 

0.1797 

0.6200 

0.2003 

199 

0.1626 

0.6530 

0.1'844 

TOO 

0.1433 

0.6800 

0.1767 

101 

0.124S 

0.7088 

0.1664 

102 

0.1 198 

0.7473 

0.1329 

103 

0.1003 

0.S009 

0.0988 

104 

0.0967 

0.8700 

0.0333 

105 

0.0967 

0.9469 

-  0.0436 

108 

0.0946 

1.0121 

-0.1067 

107 

0.0800 

1.0299 

-0.1099  . 

■  108 

0.0367 

0.9433 

0.0200 

TABLE  91. 
Values  of  the  Psychometric  Functions. 


SUBJECT  VL 


Tk 

G 

H 

L 

84 

0.0200 

0.0067 

0.9733 

85 

0.0235 

0.0171 

0.9594 

86 

0.0476 

0.0207 

0.9317 

87 

0.0800 

0.0218 

0.8982 

88 

0.1133 

0.0233 

0.8634 

89 

0.1440 

0.0272 

0.8288 

90 

0.1814 

0.0347 

0.7839 

91 

0.1962 

0.0465 

0.7573 

92 

0.2200 

0.0633 

0.7167 

93 

0.2442 

0.0853 

0.6705 

94 

0.2697 

0.1129 

0.6174 

f5 

0.2965 

0.1458 

0.5577 

96 

0.3233 

0.1867 

0.4900 

97 

0.3479 

0.2338 

0.4183 

98 

0.3670 

0.2883 

0.3447 

99 

0,3768 

0.3504 

0.2728 

100 

0.3733 

0.4200 

0.2067 

101 

0.3532 

0.4963 

0.1505 

102 

0.3148 

0.5779 

0.1073 

103 

0.2591 

0.6616 

0.0793 

104 

0.1900 

0.7433 

0.0667 

105 

0.1181 

0.8166 

0.0653 

106 

0.0599 

0.8728 

0.0673 

107 

0.0408 

0.9001 

0.0591 

108 

0.0967 

0.8833 

0.0200 

APPENDIX 


221 


TABLE  92. 


V.M.UES    OF    THE    PSYCHOMETRIC    FUNCTIONS.       SUBJECT    VII. 

I"k 

G 

H 

L 

84 

0.0133 

0.0067 

0.9800 

85 

O.llVH) 

-  0.0242 

0.9052 

86 

0.1366 

-0.0252 

0.8886 

87 

O.llSl 

-0.0119 

0.8938 

88 

0.0967 

0.0067 

0.8966 

89 

0.0903 

0.0248 

0.S849 

90 

0.1059 

"    0.0406 

0.8535 

91 

0.1422 

0.0538 

0.8040 

92 

0.1933 

0.0667 

0.7400 

93 

0.2504 

0.0819 

0.6677 

94 

0.3045 

0.1024 

0.5931 

95 

0.3474 

0.1310 

0.5216 

96 

0.3733 

0.1700 

0.4567 

97 

0.3793 

0.2205 

0.4002 

98 

0.3657 

0.2828 

0.3515 

99 

0.3362 

0.3557 

0.3081 

100 

0.2967 

0.4376 

0.2666 

101 

0.2550 

0.5223 

0.2227 

102 

0.2187 

0.6078 

0.1735 

103 

0.1941 

0.6878 

0.1181 

104 

0.1833 

0.7567 

0.0600 

105 

0.1816 

0.8086 

0.0098 

106 

0.1743 

0.8386 

-0.0129 

107 

0.1337 

0.8430 

0.0233 

108 

0.1400 

0.8200 

0.0400 

TABLE  93. 
Calculated  axo  Observed  Values  of  the  Threshold  in  the  Direction  of  Increase 


Result  of  Method  of  Just  Perceptible  Difference 

Value  found  by 

Interpolation 

Observed                                Calculated 

I 

99.60 

99.45 

100.95 

II 

98.71 

98.83 

99.55 

III 

99.58 

99.28 

100.32 

IV 

98.24 

98.08 

98.26 

V 

97.35 

97.14 

95.83 

VI 

100.33 

99.86 

101.04 

VII 

99.63 

99.86 

100.74 

TABLE  94. 
Calculated  \sa  OBSEiiVEo  Values  of  the  Th.^eshold  in  the  Direction  of  Deckease 


Subject 

Result  of  Method  of  Just  Perceptible  Difference 

Value  found  by 

Intcrpolalion 

Observed 

Calculated 

I 

93.49 

93.30 

93.26 

11 

94.98 

94.87 

95.20 

III 

97.88 

97.85 

98.65 

IV 

95.56 

95.39 

95.24 

V 

94.57 

94.47 

93.75 

VI 

95.20 

95.31 

95.82 

VII 

96.74 

95.79 

95.33 

14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

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COT  2  9 


OCT  2 1 1962 


-tOAI4 


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